Synthesis of superposition shape images by light interacting with layers of lenslets

ABSTRACT

The present invention describes methods and apparatuses for creating superposition shape images by superposed base and revealing layers of lenslet gratings. The superposition shape images form a message recognizable by a human observer or by an image acquisition and computing device such as a smartphone. The superposition shape images may be created by different superposition techniques ranging from 1D moiré, 2D moiré and level-line moiré superposition techniques to lenticular image and phase shift superposition techniques. Moiré superposition techniques enable creating superposition shape images at different apparent depth levels. Applications comprise the protection of documents and valuable articles against counterfeits, the creation of eye-catching advertisements as well as the decoration of buildings and exhibitions.

The present invention is related to the following U.S. patents:

-   (a) U.S. Pat. No. 7,194,105, filed Oct. 16, 2002, entitled    “Authentication of documents and articles by moiré patterns”,    inventors Hersch and Chosson, (category: 1D moiré);-   (b) U.S. Pat. No. 7,751,608, filed 30th of Jun., 2004 entitled    “Model-based synthesis of band moiré images for authenticating    security documents and valuable products”, inventors Hersch and    Chosson (category: 1D moiré);-   (c) U.S. Pat. No. 7,710,551, filed Feb. 9, 2006, entitled    “Model-based synthesis of band moiré images for authentication    purposes”, inventors Hersch and Chosson (category: 1D moiré);-   (d) U.S. Pat. No. 7,295,717, filed Oct. 30, 2006, “Synthesis of    superposition images for watches, valuable articles and publicity”,    inventors Hersch, Chosson, Seri and Fehr, (categories: 1D moiré and    level-line moiré);-   (e) U.S. Pat. No. 7,305,105 filed Jun. 10, 2005, entitled    “Authentication of secure items by shape level lines”, inventors    Chosson and Hersch (category: level-line moiré).-   (f) U.S. Pat. No. 6,249,588 filed Aug. 28, 1995, entitled “Method    and apparatus for authentication of documents by using the intensity    profile of moiré patterns”, inventors Amidror and Hersch (category    2D moiré).-   (g) U.S. Pat. No. 6,819,775, filed Jun. 11, 2001, entitled    “Authentication of documents and valuable articles by using moiré    intensity profiles”, inventors Amidror and Hersch (category 2D    moiré).-   (h) U.S. Pat. No. 7,058,202 filed Jun. 28, 2002, entitled    “Authentication with built-in encryption by using moiré intensity    profiles between random layers”, inventor Amidror (category 2D    random moiré).-   (i) U.S. Pat. No. 8,351,087 filed Jun. 15, 2009, entitled    “Authentication with built-in encryption by using moiré parallax    effects between fixed correlated s-random layers”, inventors Amidror    and Hersch (category random 1D and 2D moiré).

In patents (a) to (g) and (i), inventor Hersch is also inventor in thepresent patent application. Patents (a) to (i) are herein incorporatedin their entirety by reference.

BACKGROUND OF THE INVENTION

The related patents cited above provide authentication methods anddevices for preventing counterfeits of both security documents andvaluable articles and at the same time offer new means for increasingtheir attractiveness and aesthetics.

In the present application, we present a new technique for synthesizingdynamically evolving superposition shape images where the imageformation process results from the relative spatial layouts of thecontributing layers of lenslet gratings. The relative spatial layouts ofthe layers of lens gratings yield superposition shape images that mayhave a certain visual similarity with the superposition shape imagesproduced by existing layer superposition methods such as 1D-moiré,level-line moiré, phase shift methods, lenticular methods and 2D moirémethods. However, since lenslet gratings can be created at a much higherresolution than printed gratings they offer a higher protection againstcounterfeits and at the same time they allow to authenticate documentsby viewing the superposed lenslet gratings in transparency mode.

Phase Shift Techniques Phase shift effects have been used in the priorart for the authentication of documents. For example, thanks to thephase change effect, it is possible to make visible a hidden patternimage encoded within a document (see background of U.S. Pat. No.5,396,559 to McGrew, background of U.S. Pat. No. 5,901,484 to Seder,U.S. Pat. No. 5,708,717 to Alasia and U.S. Pat. No. 5,999,280 to Huang).When a revealing layer formed of a grating of transparent lines or of anarray of cylindrical lenslets is superposed on such a document, thepre-designed latent image becomes clearly visible. This phase effect hasthe particularity that the latent image does not move. When moving therevealing layer on top of the base layer, the latent image foregroundbecomes alternatively dark and bright. Phase sampling techniquescomprising screen element density, form, angle position, size andfrequency variations are described in U.S. Pat. No. 6,104,812 to Koltaiet. al. McCarthy and Swiegers teach in U.S. Pat. No. 7,916,343 that byapplying a vertical phase shift on a horizontal line grating accordingto the darkness of an original image, one creates a modified gratingpotentially capable of hiding the latent image. The latent image isrevealed by superposing the original grating on top of the modifiedgrating. 1D-moiré techniques (mainly U.S. Pat. Nos. 7,751,608 and7,710,551) 1D-moiré synthesizing methods, also called band moiré imagesynthesizing methods are characterized by equations that relate a baselayer layout comprising base bands made of vertically compressedinstances of a 1D moiré image, a revealing layer layout comprising agrating of sampling lines and the 1D moiré layout resulting from thesuperposition of the base and revealing layers. The 1D moiré imageshapes are a geometric transformation of the shapes embedded within eachband of the base band grating. This geometric transformation comprisesalways an enlargement in one dimension, and possibly a rotation, ashearing, a mirroring, and/or a bending transformation. 1D-moirésynthesizing methods enable creating a base band grating and a revealingline grating that yield upon translation or rotation of the samplingposition of the revealing line grating on the base band grating adisplacement of the 1D moiré image shapes.Shape Level Line Moiré Synthesizing Techniques (Mainly U.S. Pat. No.7,305,105)

Shape level line moiré s occur in a superposition image when a baselayer comprising a line grating locally shifted according to theelevation of a spatially laid out shape elevation profile is superposedwith a revealing layer comprising the unshifted sampling line grating.The layer with the locally shifted line grating embeds the shapeelevation profile generated from an initial, preferably bilevel motifshape image (e.g. typographic characters, words of text, symbols, logo,ornament). By modifying the relative superposition phase of the samplingrevealing layer grating in superposition with the base layer (e.g. by atranslation or rotation), one may observe as shape level line moirésuccessions of level lines of the shape elevation profile evolvingdynamically between the initial motif shape boundaries (shape borders)and shape foreground centers, respectively shape background centers,thereby growing and shrinking. The movement of shape level lines acrossthe motif shape creates visually attractive pulsing motif shapes, forexample pulsing symbols such as a pulsing heart. Shape level linetechniques have also been published in December 2014 in “S. Chosson, R.D. Hersch, Beating Shapes Relying on Moiré Level Lines, ACM Transactionson Graphics (TOG), Vol. 34 No. 1, November 2014, Article No. 9, 1-10.

Lenticular Image Synthesizing Techniques

Lenticular image synthesizing methods are well described in thebackground sections of U.S. Pat. No. 8,284,452 to Blum, U.S. Pat. No.7,255,979 to Weiss and Pilosso, U.S. Pat. No. 5,924,870 to Brosh andGottfried and U.S. Pat. No. 5,519,794 to Sandor and Meyers. A lenticularimage consists of an ordered sequence, matched to a lenticularfrequency, of a plurality of images broken down into bands or strips,which are viewed through an array of cylindrical lenslets (lenticularlenses). The period of the grating of cylindrical lenslets is equal tothe strip width multiplied by the number of the contributing of images.

Let us call the phase-shift techniques, 1D moiré techniques, shape levelline moiré techniques and lenticular image synthesizing techniques“one-dimensional line-oriented” layer superposition techniques. Let uscall 2D periodic moiré or 2D random moiré synthesizing techniques“two-dimensional” superposition techniques.

2D Moiré Techniques

2D moiré techniques are based on the moiré intensity profile that isgenerated by the superposition of a specially designed 2D base layerdot-screen and a revealing layer formed of a 2D array of transparentdots or of spherical microlenses (see U.S. Pat. No. 6,249,588 to Amidrorand Hersch, filed Aug. 28, 1995). The base layer dot-screen consists ofa lattice of tiny dots, and is characterized by three parameters: itsrepetition frequency, its orientation, and its dot shapes. When therevealing layer is laid on top of the base layer dot-screen, when bothof them have been designed in accordance with 2D moiré layouttechniques, there appears in the superposition a highly visiblerepetitive moiré pattern of a predefined intensity profile shape, whosesize, location and orientation gradually vary as the superposed layersare rotated or shifted on top of each other. As an example, thisrepetitive moiré pattern may comprise any predefined letters, digits orother symbols (such as the country emblem, the currency, etc.). The baselayer dot-screen may include dots of gradually varying sizes and shapes,and can be incorporated (or dissimulated) within a variable intensityhalftoned image such as a portrait, landscape, or decorative motif,which is generally different from the motif generated by the moiréeffect in the superposition. Embodiments of 2D moiré techniques includea revealing array of microlenses superposed with base layer imagesformed of the combination of antireflection and partially reflectingstructures (see U.S. Pat. No. 8,027,093, filed Oct. 4, 2010, inventorsCommander et al.). They also include as base layer a planar array ofimage icons and as revealing layer a planar array of image iconfocussing elements (see U.S. Pat. No. 7,333,268, filed Nov. 22, 2004,inventors Steenblik et al.).

Random Moiré 2D and 1D Techniques

U.S. Pat. No. 7,058,202 to Amidror teaches that the superposition of twospecially designed correlated random or pseudorandom 2D dot-screensyields a single instance of a moiré intensity profile which consists ofsingle instance of the moiré shape whose size, location and orientationgradually vary as the superposed layers are rotated or shifted on top ofeach other. U.S. Pat. No. 8,351,087 to Amidror and Hersch teaches acompound layer that displays a dynamically moving single moiré shapeinstance. This compound layer is formed of the superposition of a baselayer and a revealing layer with a gap between them. The layer elementsare laid out at s-random locations, the s-random locations of therevealing layer elements being derived from the s-random locations ofthe base layer elements. The base layer element locations and therevealing layer element locations are therefore strongly correlated. Thes-random locations are determined by applying pseudo-randomperturbations or displacements to a periodic set of locations. Whentilting the compound layer, the superposition of said s-random base andrevealing layers yields a single moiré shape instance, that dynamicallyvaries in its size or orientation and/or moves along a trajectorydetermined by the respective layouts of the base and revealing layers.Layouts are available in which the moiré shape moves along a directionsubstantially perpendicular to the tilting direction. The base layer mayform a halftone image by having its elements large in dark areas andthin in bright areas. It is possible to conceive a moiré shape that isburied and hidden within background random noise, so that it is notvisible when the compound layer is not tilted, and it only appears andbecomes visible upon tilting the compound layer.

Stereoscopic Depth Perception of Moiré

Elements of theory about stereoscopic vision can be found in the paperby E. Hibbard et al., “On the Theory and Application of Stereographicsin Scientific Visualization”, published in the book “From objectmodelling to advanced visual communication” edited by S. Coquillard, W.Strasser and P. Stucki, Springer Verlag (2004), pp 178-196. The paper“The moiré magnifier” by M C Hutley, R Hunt, R F Stevens and P Savanderpublished in “Pure and Applied Optics: Journal of the European OpticalSociety Part A Vol. 3 No 2, pp 133-142 already points to the possibilitythat moiré effects can be seen in stereoscopic vision. The paper by J.Huck, “Moiré patterns and the illusion of depth”, published at the Intl.Conf. of the International Society of Arts, Mathematics and Architecture(ISAMA), June 2004 indicates how to compute the position and period ofthe moiré light intensity profile resulting from two vertical layers ofvertical straight line gratings separated by a given gap and illuminatedfrom behind. U.S. Pat. No. 7,333,268 to R. A. Steenblick, M. J. Hurt andG. R. Jordan describes for the case of 2D moiré s when a moiré is infront and when a moiré is in the back of the superposed 2D layers. Inthe present disclosure, we show how to calculate and synthesize 1D moiréshapes having a desired perceived depth when viewed stereoscopically bya human.

Prior Art Microlens and Lenticular Lens Superposition Methods

U.S. Pat. No. 7,931,305 to Tompkin and Schilling teaches the creation ofa transparent window incorporating microlens fields on both sides of thewindow. The system may behave as an individual macroscopic lens.Depending on parameters such as lens spacing and lens diameter, variousoptical effects are obtained. Items of information may be obtained byhaving different regions with different lens spacing parameters.Optically, these different regions become apparent to the viewer. Incontrast to the present invention, U.S. Pat. No. 7,931,305 does notallow to conceive predefined superposition images having a predefineddynamic behavior, such as moving moiré shapes, shapes with level linestravelling from their center to their borders and vice-versa ordynamically moving shapes formed of successively visible shapeinstances.

U.S. Pat. No. 8,705,175 B1 to Lundgen and Sarda, filed Mar. 14, 2013,priority Apr. 11, 2012, teaches a method of producing a two-sidedlenticular film that exhibits an illusion of stripes embedded within thefilm.

Prior Art Supersposition Image Synthesizing Techniques

In the prior art, phase-shift techniques, 1D or 2D moiré techniques,either repetitive or random, shape level line moiré techniques andlenticular image synthesizing techniques assume that the base layerinformation is printed or patterned into the base layer alonglongitudinal 1-dimensional structures such as bands or as 2-dimensionalarray structures and that a revealing layer is made of a line-oriented1-dimensional array or respectively of a 2-dimensional array samplingthe base layer. This sampling revealing layer is made of transparentlines or of cylindrical lenslets (lenticular lenses) for the 1D case orof substantially spherical lenses for the 2D case. In phase shifttechniques, the base layer information comprises, at given locations,base layer structures shifted by a fraction of the revealing layersampling line period. In 1D moiré techniques, the base layer informationcomprises the base bands, each base band incorporating base band shapesobtained by a linear or non-linear geometric transformation of thedesired 1D moiré shapes. In 2D moiré techniques, the base layerinformation comprises juxtaposed dot areas containing dot shapesobtained by a linear or non-linear geometric transformation of thedesired 2D moiré shapes. In shape level line moiré techniques, the baselayer information comprises a line grating or a grating of dither bandslocally shifted in proportion to the elevation profile at the currentposition. In lenticular image synthesizing techniques, the base layerinformation comprises the bands representing sections of thecontributing images. Embodiments include the creation of a compound madeof the revealing layer on one side and of the base layer on the otherside of a substrate having a given thickness. When tilting thiscompound, the revealing layer sampling elements sample different partsof the base layer bands and the superposition image evolves dynamically,according to the implemented superposition image synthesizing technique.

In the present disclosure, we propose for both one-dimensionalline-oriented and for two-dimensional layer superposition techniques,repetitive or random, to replace the base layer printing or patterningpresented in the prior art by the placement of one-dimensional lightconcentrating lenslets (e.g. cylindrical lenslets) in the backgroundareas of the base layer shapes. Base layer lenslets may be created onone side of a substrate by a roll-to roll-process simultaneously withthe creation of the revealing layer sampling lenslets on the other sideof the substrate, thus avoiding shift and rotational inaccuraciesbetween the base and revealing layers.

SUMMARY OF THE INVENTION

The present invention aims at creating a superposition shape image thatshows a recognizable message with the superposition of a base layercomprising lenslet gratings and a revealing layer comprising a lensletgrating. The superposition shape image is created with a superpositiontechnique selected from the set of 1D moiré s, 2D moiré s, random moiré,level line moiré, lenticular image, phase shift and stereoscopic depthsynthesizing techniques. Each superposition technique has its ownmathematical basis relating the revealing layer grating layoutparameters to the base layer grating layout parameters, especially therevealing layer period and orientation and the base layer period andorientation. Depending on the considered superposition technique, therevealing layer is either formed of a 1D grating of substantiallycylindrical lenslets or by a 2D grating of substantially sphericallenslets. The base layer comprises foreground and background shapesderived from the foreground and background of the superposition shapeimage. For example, in case of a 1D moiré, the base layer shapes are ageometrical transformation of the superposition shape image formed bythe moiré.

In order to create superposable revealing and base layer lensletgratings, one needs to determine the position of the individual surfacesdefining the layout of the lenslet gratings according to the selectedsuperposition technique and the desired superposition shape image, forboth the base and the revealing layers. With the surfaces specifying thelayout of the lenslet gratings, it becomes possible to fabricate thelenslet gratings by applying techniques such as lithography, laserwriting, etching, reflow and embossing.

In case that the base and revealing layer lenslet gratings form a fixedsetup, and when the setup is illuminated from behind or is shown infront of light reflecting surface, tilting the setup yields a visibledynamically evolving superposition shape image that is easy torecognize. In case of a 1D or 2D moiré, the dynamically evolvingsuperposition shape image is characterized mainly by a displacement. Incase of a level line moiré, it is characterized by lines of constantintensity or color laid out along the level lines of the elevationprofile of the superposition shape. These constant intensity or colorlines evolve across successive level lines between the superpositionshape boundaries and the shape foreground and background centers. Incase of a lenticular image, the dynamically evolving superposition shapeis formed of a succession of related sub-images and in case of a phaseshift superposition technique it is formed by an inversion of intensityor by a switch between colors.

In order to provide an additional protection against counterfeits, it isalso possible to apply geometrical transformations to both the base andthe revealing layers. This yields revealing and base layer shapes havinga curvilinear layout. In case of a 1D moiré, the base layer is generatedaccording to a geometric transformation derived from the specifictransformation of the revealing layer and a desired layout of the 1Dmoiré expressed by a corresponding moiré layer geometric transformation.In case of a level line moiré having the same appearance as the levelline moiré created with a rectilinear revealing layer, the base layer isgenerated according to the same specific geometric transformation as therevealing layer and then the elevation profile is incorporated byvertical shifts of the base layer surfaces proportional to the elevationprofile. In case of a curvilinear level line moiré being geometricallytransformed according to the specific transformation of the revealinglayer, the base layer is first shifted in proportion to the elevationprofile and then generated according to the same specific transformationas the revealing layer.

In the case of a level line moiré, the array of revealing layer surfacesspecifying the layout of the revealing layer lenslet array is an arrayof revealing layer transparent lines. The arrays of surfaces forming thebase layer foreground shapes specifying the layout of the base layerlenslet gratings are arrays of base layer transparent lines, arrays ofrectangles or arrays of disks. In the case of base layer transparentlines, the fabricated base layer lenslet gratings have substantially thesame period as the fabricated revealing layer lenslet grating. In thecase of arrays of rectangles or arrays of disks, the fabricated lensletgratings have a substantially smaller period compared with the period ofthe revealing layer lenslet grating. The base layer background may beleft without lenslet gratings or filled with randomly positionedmicrolenses of sizes that are randomly selected within a given sizeinterval, and are substantially smaller than the period of the revealinglayer grating.

In the case of superposed base and revealing layer lenslet gratings thatform a fixed setup, with the revealing layer lenslet grating having avertical orientation, the eyes of an observer see different views of thebase lenslet gratings. These different views create a parallax effectallowing to perceive the superposition shape image as an image composedof shapes having different apparent depths. The superposition shapeimage may form two messages, one at a certain depth level and the secondone at a different depth level. When tilting the setup, the messages maymove in inverse directions and at different apparent depth levels.

One may also create base layer lenslet gratings that when viewed aloneshow a halftone image and when viewed in superposition with therevealing layer show a visible and recognizable message enabling toauthenticate the base layer. The halftone image may be formed of anyvariable intensity image such as landscapes, flags, vehicles, faces,persons, dresses, luxury articles, watches, fruits, trees, logos,instruments, utility objects, planes, rockets, weapons, etc.

In the case of a level line moiré, when the illumination of the fixedsetup comprises spatially varying colors, the level lines will havecolors that are similar to the colors present in the illumination. Theillumination with the different colors may be realized with a largedisplay, with colored bulbs or with colored light emitting diodes(LEDs). As a decorative feature, one may include several LEDs within aled package. By driving the LEDs individually, i.e. by having anexecutable program setting their respective emission intensities and byvarying these intensities, one may create level line moiré s with colorsthat evolve across the color space at successive time intervals.

On a setup formed of superposed base layer lenslet gratings and of arevealing layer lenslet grating, the superposition shapes form therecognizable message. These superposition shapes are formed by thesampling action of the revealing layer lenslet grating on the plane onwhich the base layer lenslet gratings concentrate the incoming light.The recognizable message moves dynamically when changing the observationangle or the observation location in respect to the superposed lensletgratings. The recognizable message can be formed of text, numbers,graphical symbols, typographical characters, numerals, logos, andspatial codes such as barcodes and QR codes.

A smartphone, tablet or laptop computer may capture the superpositionshapes forming a visible message and verify its authenticity withauthentication software operable for recognizing the message and forcomparing its signature with signatures located in its memory, or bysending the visible message or its signature to a remote server locatedon the Internet and receiving a reply indicating whether the visiblemessage is authentic or not.

An apparatus for producing superposable revealing layer grating and baselayer lenslet gratings that show superposition shapes forming arecognizable message comprises a computer with a software moduleinteracting with the user, interacting with other computers or readinginstructions from a file in order to select a superposition techniquefrom the set of 1D moiré s, 2D moiré s, random moiré, level line moiré,lenticular image, phase shift and stereoscopic depth synthesizingtechniques. On this computer, the same or a different software module isoperable for synthesizing the layout of the base layer lenslet gratingsand the layout of the revealing layer lenslet grating according to theselected superposition technique. The apparatus further comprises meansto expose and develop resist structures laid out according to the layoutof the lenslet gratings, heating means operable to apply a reflowprocess to the exposed and developed resist structures, means to createmolds containing the negatives of the reflowed resist structures, aroll-to-roll device incorporating the molds to create the lensletgratings, UV curable material pressed by the roll-to-roll device intothe molds, UV illumination means operable to cure the material in themolds and possibly a system to cut and eject the cured material formingthe lenslet gratings.

In case a fixed setup of base and revealing layer lenslet gratings is tobe produced, one roll-to-roll device creates the base layer lensletgratings on one side of a substrate and a second roll-to-roll devicecreates the revealing layer lenslet grating on the other side of thesubstrate, in registration with the base layer lenslet gratings. As analternative, a single roll-to-roll device may create at the same timethe base layer lenslet gratings on one side of a substantiallytransparent substrate and the revealing layer gratings on the other sideof the substrate at the same location.

Optionally, an additional polymer having an index of refraction lowerthan the one of the lenslet gratings may be deposited and hardened ontop of the cured material forming the lenslet grating. This additionalpolymer creates a flat surface. This can be carried out both for thebase and revealing layer lenslet gratings. Then, one may create a fixedsetup looking like a flat piece of plastic, but capable of showingdynamically evolving superposition shapes.

Further fabrication methods comprise polymer jetting devices workinglike ink jet printers, possibly located into closed enclosures enablingprogrammable heating and UV curing. For large size setups of lensletgratings, it is also possible to directly print the base and revealinglayer lenslet gratings by describing them as 3D surface models,converting the surface description into 3D printer head movements andprinting these models with a substantially transparent plastic material.Such medium to large size setups of lenslet gratings have a highdecorative value and may be used for luxury articles, advertisement,exhibitions and in amusement parks.

The proposed superpositions of revealing and base layers of lensletgratings offer a strong protection against counterfeits, since thesegratings cannot be produced without sophisticated equipment allowingprecise lithography and reflow operations. Moiré superpositiontechniques are very sensitive to small deviations in layout andsuperposition. Therefore, a superposition shape image forming arecognizable message cannot be reproduced by counterfeiters withoutintroducing serious deformations. In addition, the revealing layergrating of lenslets may have a curved layout such as a cosinusoidallayout. Without knowing the parameters of the corresponding layout,faithful reproduction is extremely difficult and time-consuming.Finally, one or both layers of lenslet gratings may each be encapsulatedby a transparent material layer such as a polymer having a lower indexof refraction than the index of the lenslets. The encapsulating layerhas a flat interface with the air and hides therefore the layout of theencapsulated base lenslet grating. Such an encapsulation makes it verydifficult for a counterfeiter to recover the orientation, size andlayout of the lenslet gratings. A unauthorized replication of a setupcomprising encapsulated base and revealing layer lenslet gratings istherefore extremely difficult to achieve.

The shape image created by superposed layers of lenslet gratings forms arecognizable message that dynamically evolves in synchronization withthe movement of an observer. Since it is the movement of the humanobserver's eyes that drives the evolution of the message, there is animmediate feedback. Such a feedback is unusual and strongly attracts theattention of the observer. Several persons may simultaneously observethe superposed layers of lenslet gratings. Every person will see from adifferent spatial position a slightly different instance of thedynamically evolving message.

In addition to providing a protection against counterfeits, thepresented fixed setups of revealing and base layer lenslet gratingsyield superposition shape images that have a high esthetical anddecorative value and may also be attractive for luxury products such aswatches, smartphones, perfumes, expensive drinks, for clothes such as adress, a skirt, a blouse, a jacket, shawls and pants as well as in bikesand cars. Superposed revealing and base layer lenslet gratings may alsobe used for advertisement, for the decoration of buildings, for showingsurprising messages on exhibition walls, and in amusement parks.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a multi-lenslet imaging setup made of a revealing layer 100with a cylindrical lenslet grating having a large repetition period anda base layer 110 whose foreground base layer shapes are filled with agrating of cylindrical lenslets 102 having a small period;

FIG. 2 shows a cross-section through the revealing and base layergratings of lenslets having respectively large and small replicationperiods;

FIG. 3 shows a cross-section through revealing and base layer gratingsof lenslets having the same replication periods;

FIG. 4A shows a base layer where the arrays of rectangles indicate theforeground area of the base layer shapes on which cylindrical lensletgratings will be placed and where the background is left empty;

FIG. 4B shows the same base layer as FIG. 4A, but with the produced baselayer lenslet gratings 404, 405, 406 at the positions of the rectanglearrays and with the background filled with diffusing microlenses 403;

FIG. 4C shows an enlarged view of FIG. 4B;

FIG. 5A shows for the 1D moiré the base layer formed of base bands 501and the revealing layer sampling lines 502 a, 502 b, 502 c as well asthe resulting moiré shape 503;

FIG. 5B shows the succession of 1D moiré shapes 503 obtained by thesuperposition of the base and the revealing layers;

FIG. 6A shows a rectilinear 1D moiré obtained by superposition of thebase band layer and the revealing layer;

FIG. 6B shows a part of the base layer 610 populated by small obliquerectangles defining the layout of the cylindrical lenslets;

FIG. 6C shows a view under the microscope of the same area as in FIG.6B, after fabrication of the base layer lenslet gratings;

FIG. 7A shows another embodiment of base layer area 610 in FIG. 6A, withthe rectangle array 712 defining the layout of a horizontally laid outlenslet grating 720;

FIG. 7B shows under the microscope rectangular area 701 (dashed border)embodied by a horizontal lenslet grating 720 for the base layerforeground area and a vertical lenslet grating 721 for the base layerbackground area;

FIG. 8 shows a photograph of the fixed setup of base and revealing layerlenslets, with the revealing layer lenset grating 801 and the resultingmoiré shape image “EPFL” 802;

FIG. 9 shows the same device as in FIG. 8, but after tilting itvertically: the moiré shapes have moved to a lower position 902;

FIG. 10A shows a circularly laid out 1D moiré image obtained bysuperposition of a geometrically transformed base layer and arectilinear revealing layer;

FIG. 10B shows a photograph of the view from a microscope of a part ofthe fixed setup comprising base and revealing layer lenslet gratings,fabricated according to the layout of the base and revealing layers ofFIG. 10A;

FIG. 11 shows a photograph of the fixed setup comprising base andrevealing layer lenslet gratings laid out according to FIG. 10A,yielding the desired circular moiré shapes;

FIG. 12 shows a circularly laid out 1D moiré image obtained bysuperposition of a geometrically transformed base layer and acosinusoidal geometrically transformed revealing layer;

FIG. 13 shows the same base and revealing layers as FIG. 12, but withthe revealing layer sampling a different position of the base layer,yielding a radial displacement of the circular moiré shapes;

FIG. 14 shows the geometry used for calculating the offset between themoiré shape seen by the left eye and the moiré shape seen by the righteye;

FIG. 15 shows schematic views of the base layer 151, of the moiré shapeseen by the left eye 152 and of the moiré shape seen by the right eye153, with the apparent revealing layer period being larger than the baselayer period;

FIG. 16 shows similar representations as in FIG. 15, but with the baselayer period being larger than the apparent revealing layer period;

FIGS. 17 and 18 are helpful in calculating the apparent depth of themoiré shape view;

FIGS. 19A and 19B show the same setup of base and revealing layerlenslet gratings at two different horizontal tilt angles, yieldingdifferent relative positions of the “VALID” and the “OK” moiré shapes;

FIG. 20A shows a base layer formed of a 2D array of micro-shapes sampledby a revealing layer made of a 2D array of microholes yielding oneinstance of a 2D array of “$” moiré shapes;

FIG. 20B shows a photograph of an enlarged portion of a realization ofthe base and revealing layers shown in FIG. 20A, with the foreground ofthe micro-shapes covered with gratings of small cylindrical lenslets andwith their background covered with randomly placed small sphericalmicrolenses;

FIG. 21 shows schematically a detailed view of the revealing layerlenses sampling the base layer micro-shapes that are embodied withgratings of cylindrical lenslets concentrating the light from behind;

FIG. 22 shows a photograph of a 2D moiré setup formed of base andrevealing layer gratings of lenslets;

FIG. 23 shows the layout of a desired moiré shape array and FIG. 24shows the corresponding layout of the base layer micro-shape array;

FIG. 25A shows base and revealing layer rectilinear thick line gratingsand FIG. 25B shows the same gratings but geometrically transformed;

FIG. 26A gives an example of bilevel shapes;

FIG. 26B shows the boundaries, foreground and background skeletons ofthese bilevel shapes;

FIG. 26C shows the corresponding computed elevation profile;

FIG. 27A shows a base layer made of gray bands shifted verticallyaccording to the elevation profile shown in FIG. 26C;

FIG. 27B shows the level line moiré shapes obtained by superposing asampling revealing layer on top of the vertically shifted gray bands;

FIG. 28A shows a photograph of the setup obtained by laying out thecylindrical lenslets into the base layer along the same paths as thegray bands and by replacing the rectilinear revealing layer samplingline array with a grating of rectilinear cylindrical lenslets of thesame period;

FIG. 28B shows an enlargement of the black square region 2808 of FIG.28A;

FIGS. 29A, 29B, 29C and 29D show how the relative positions and the tiltangle of the base and revealing lenslet gratings influence the lightintensity reaching the observer;

FIGS. 30A, 30B, 30C and 30D show a geometrically transformed base layerlayout, a corresponding geometrically transformed revealing layerlayout, the superposition of the two layers at a first sampling phaselocation and the superposition of the two layers at a second samplingphase location, respectively;

FIGS. 31A and 31B show at two different tilt angles an example of alevel line moiré shape produced with superposed base and revealinglayers of cylindrical lenslet gratings, the base layer lenslet gratingsbeing shifted according to the intensity of the face that is to appearas level-line moiré;

FIG. 32A shows an original grayscale image;

FIG. 32B shows the corresponding halftoned image with substantiallyparallelogram-shaped black halftone screen elements embedding theelevation profiles shown in FIGS. 33A and 33B;

FIG. 33A shows a first elevation profile containing identifyinginformation laid out along a small positive slope;

FIG. 33B shows a second elevation profile with identifying informationlaid out along a small negative slope;

FIGS. 33C and 33D show an array of dither bands shifted perpendicularlyto their orientation according to a small portion 3301 of the elevationprofile of FIG. 33A and an array of dither bands shifted perpendicularlyto their orientation according to a small portion 3302 of the elevationprofile of FIG. 33B, respectively;

FIG. 33E shows an enlargement of the combination of the two elevationprofiles, by selecting at each position the lowest intensity and byperforming an histogram equalization;

FIG. 33F shows an enlarged portion (square 3201) of the halftoned imageshown in FIG. 32B, obtained by dithering the original image in FIG. 32Awith the elevation profile shown in FIG. 33E;

FIG. 34A shows the small rectangles specifying the layout of thecylindrical lenslet gratings within the surfaces of the blackparallelograms in the halftoned image shown partly in FIG. 33F;

FIG. 34B shows an enlargement of FIG. 33A;

FIG. 35A shows a portion of the moiré identifying information appearingwhen superposing the halftoned image 3501 corresponding to the eye andnose 3202 in FIG. 32B with a revealing sampling layer 3502 having thesame orientation as the array of dither bands shown in FIG. 33C;

FIG. 35B shows a portion of the moiré identifying information appearingwhen superposing the same portion 3202 of the halftoned image 3501 as inFIG. 35A with a revealing sampling layer 3503 having the same baseorientation as the array of dither bands shown in FIG. 33D;

FIG. 36 shows how a lenticular base layer image is constructed from fourdifferent parts;

FIGS. 37A, 37B, 37C, 37D show photographs of a lenticular image setupcomprising base and revealing layer lenslet gratings that displays asuccession of four successive lenticular shape images upon verticaltilting of the setup;

FIG. 38 shows fabrication steps for realizing gratings of lenslets;

FIG. 39 shows a roll-to-roll mechanism enabling producing lensletgratings efficiently;

FIG. 40 shows a second roll-to-roll mechanism enabling producing lensletgratings on two sides of a substrate;

FIG. 41 shows a production line for producing personalized gratings oflenslets with a polymer jetting print engine;

FIG. 42 shows the steps for producing with a 3D printer personalizedsuperposable base and revealing layer gratings;

FIG. 43 shows the preparation steps necessary before fabricating thesuperposed base and revealing layers of gratings;

FIGS. 44A and 44B show a setup 4400 with base layer lenslet gratings ontop of a semi-transparent substrate and a revealing layer lensletgrating behind that substrate, that, when viewed in reflecting modeshows a halftone image (FIG. 44A) and when viewed in transmissive modereveals a hidden information (FIG. 44B);

FIGS. 45A and 45B show a setup formed of base and revealing layerlenslet gratings that upon tilting yields a displacement of theappearing moiré shapes;

FIG. 46 shows a smartphone capturing a shape image from the setupcomprising base and revealing layer lenslet gratings, recognizing themessage in that shape image and possibly in connection with a remotehost computer validating or not the message present in the capturedshape image.

DETAILED DESCRIPTION OF THE INVENTION

The superposition images generated by the phase-shift techniques, 1D and2D moiré techniques, shape level line moiré techniques and lenticularimage synthesizing techniques result from sampling a base layercomprising foreground and background shapes by a revealing layer made ofan array of lenslets. The present invention aims at replacing the baselayer printing or patterning techniques used for producing the baselayer foreground and background shapes known from the prior art bypopulating the foreground or background areas of the base layer withsubstantially cylindrical lenslet gratings.

For the phase-shift techniques, the 1D moiré techniques, a category ofshape level-line moiré embodiments and the lenticular image synthesizingtechniques, the cylindrical base layer lenslets have a substantiallysmaller replication period than the replication period of thecylindrical lenslets forming the 1-dimensional revealing layer samplinglenslet grating. For the 2D moiré technique, the base layer cylindricallenslets have a substantially smaller replication period than the periodof the revealing layer 2-dimensional grating of spherical samplinglenslets.

For another category of shape level line moiré embodiments, the baselayer cylindrical lenslets should have substantially the samereplication period as the revealing layer grating of cylindricallenslets.

Vocabulary

In the present application, we use the term “cylindrical lenslets” or“1D lenslets” as a generic term for lenslets whose cross-section aree.g. a section of a circular disk or a section of a parabola and thatfollow straight or curvilinear lines. A grating of cylindrical lensletsmay cover a region of the plane. Between each lenslet of a grating ofcylindrical lenslets, there may be no space or a small space. The periodof such a grating is defined as the repetition period its cylindricallenslets. Gratings of cylindrical lenslets are often fabricated from adescription of longitudinal rectangles by applying lithographic andreflow techniques. A “longitudinal rectangle” is defined as a relativelylong and thin quadrilateral having a constant width. Cylindricallenslets following a long straight or curvilinear path are fabricatedfrom a description of “longitudinal stripes” of constant width.

We use the term “spherical lenslets”, “spherical lenslet grating” or “2Dlenslet grating” as a generic term for lenslets that may fill the spacein a repetitive 2D manner, e.g. as a regular 2D array. Their shape maybe spherical, aspherical or partly spherical and partly non-spherical.

The term “multi-lenslet setup” refers to a fixed setup comprisingsuperposed base and revealing layer gratings. In general, the revealinglayer grating (FIG. 17, 143) and the base layer gratings (FIG. 17, 174)are parallel and have in common a plane on which the revealing layergrating samples the light concentrated by the base layer lensletgratings (FIG. 3, 303, FIG. 17, 147).

In general, we use the term “revealing layer lenslet grating” in thesingular form for the revealing layer (e.g. FIG. 8, 801) and “base layerlenslet gratings” in the plural form because the base layer comprisesmany base layer micro-shapes (e.g. FIG. 4B, 404, 405, 406) that are eachcovered by a lenslet grating. However, it is possible to have severalrevealing layer lenslet gratings on the same revealing layer.

We use the term “recognizable message” for the message that is generatedas a superposition shape image by the superposed base and revealinglayer lenslet gratings. “Recognizable” means that either a human beingor a computing system is capable of recognizing the message, being it apicture with recognizable elements such as a flag, a face, a house, aforest, a horse, be it a string of letters such as a number or a codeformed of numbers and letters, be it a 1D or 2D barcode, or be it aQR-code recognizable by a computer or a smartphone.

We use the term smartphone for a computing device incorporating a cameraand being connected to a server for information exchange. Commerciallyavailable tablet or laptop computers may also perform the same actionsas the smartphone.

Multi-Lenslet Setup with Large Revealing Layer Lenslet Periods and SmallBase Layer Lenslet Periods

FIG. 1 illustrates a multi-lenslet setup 100 comprising the revealinglayer cylindrical lenslet grating 101 on top of a substantiallytransparent substrate 103. On the other side of the transparentsubstrate, the base layer comprises a cylindrical lenslet grating 102forming the foreground of a vertically compressed letter “E” (110). Thebase layer lenslets have their cylindrical parts on the back side (102,107) of the setup. This setup may be observed by looking 106 from thefront side of the revealing layer lenslet grating 101. The incominglight 105 irradiates the setup from the backside, i.e. from the baselayer lenslet gratings. Part 100 of the figure represents themulti-lenslet imaging setup, part 110 an enlargement of one of thereplicated lenslet gratings forming a base layer base band (see Section“Creating 1D moiré with the multi-lenslet setup) and part 120 anenlargement of a part of the base layer grating comprising twocylindrical lenslets.

FIG. 2 illustrates a cross-section 201 through either a cylindrical lens(for line oriented superposition effects) or a spherical lens (for a 2Dmoiré) being part of the revealing layer. This revealing layer issuperposed with base layer lenslet gratings 208 shown as cross-sectionsacross the base layer 202. The plane 203 is the focal plane of therevealing layer lenslets on which the base layer cylindrical lensletsconcentrate the incident light. The revealing layer cylindrical orspherical lenslet is repeated at a period T_(r) (206). Its width isW_(r) (207) and its focal length is ƒ_(r) (208). In the present example,the cylindrical lenslets 213 forming the base layer 202 cover the baselayer foreground shape 208 and do not cover the base layer backgroundshape 209. These lenslets have a period T_(bl) and have a focal lengthf_(b) which, in the case of a 1D moiré, 2D moiré and of lenticularimages, are considerably smaller than the revealing layer cylindricallens period T_(r) and focal length ƒ_(r), respectively. Between therevealing layer lenslets 201 and the base layer with or without lenslets202 there is a substantially transparent substrate whose thickness h_(s)(212) is related to the sum of the focal lengths of the revealing andthe base layer lenslets minus the lens heights e_(r) and e_(b) (see FIG.2), i.e. h_(s)≈ƒ_(r)+ƒ_(b)−e_(r)−e_(b).

The observer located at a normal viewing distance from the revealinglayer may view the multi-lenslet imaging device formed of the revealingand base layers from one angle (e.g. perpendicularly, see 210) or fromanother angle (e.g. angle α, see 211). By tilting this multi-lensletimaging device, the revealing layer lenslets sample the base layereither at a location where the base layer lenslets are present or at alocation where they are absent. Regions with lenslets create a brighttexture differentiating themselves from the regions without lenslets.This differentiation is at the base of the superposition images viewedby the observer. When tilting the device in respect to the observer, therevealing layer lenslets scan their focal plane 203 close to the baselayer, thereby propagating the light intensities created by the presenceor the absence of the base layer lenslets onto the observer's eyes.

Multi-Lenslet Setup with Similar Revealing Layer and Base Layer Periods.

The superposition of an array of revealing layer lenslets having a largeperiod and of an array of base layer lenslets having a small period isadequate for the 1D and the 2D moiré synthesizing methods, for some ofthe level line methods, for some of the phase shift synthesizing methodsand for the lenticular image synthesizing methods. In respect to someother level line moiré and phase shift synthesizing technologies, therevealing layer cylindrical lenslets and the base layer cylindricallenslets shall have the same period, or an integer multiple of thatperiod, but are in some portions of the superposition image shifted onein respect to the other.

FIG. 3. shows schematically a multi-lenslet imaging setup where therevealing layer lenslets have a period T_(r) (330) and a focal lengthƒ_(r) (308) and where the base layer lenslets have the same periodT_(b)=T_(r) (331) and a focal length ƒ_(b). The focal plane 303 isdefined by the focal length of the revealing layer lenslets. The baselayer lenslets concentrate the light on the focal plane of the revealinglayer lenslets. The base layer lenslets need not necessarily focus thelight on the focal plane sampled by the revealing layer lenslets. Theirdistance 309 to the focal plane can be different from ƒ_(b) by up to±20%. A simple concentration of the incident light is sufficient. Theobserver 335 is located at a certain distance from the front of thesetup, typically 35 cm, when the setup is located on a security documentor a valuable article.

A strong visual effect is obtained by illuminating the multi-lensletimaging device with spatially varying intensities or colors, for examplea display or LEDs (light emitting diodes) showing red 354, green 355,blue 356 and white 357 colors. The corresponding light rays 314, 315,316 and 317 concentrated by the base lenslet grating 302 illuminate inthe focal plane 303 of the revealing layer lenslets portions 364 a, 365a, 366 a, 367 a for lenslet 302 a, portions 364 b, 365 b, 366 b, 367 bfor lenslet 302 b, portions 364 c, 365 c, 366 c, 367 c for lenslet 302 cetc. . . . . Instead of these 4 distinct colors, continuous ornon-continuous intensity and/or color variations also create a strongvisual effect.

In phase shift and level line moiré methods, some of the base layerlenslets are shifted in respect to the revealing lenslets. For example,the base layer lenslet 302 a with center at position 323 is in phasewith the revealing layer lenslet 301 a with center at position 320. Butthe base layer lenslet 302 b with center at position 324 is shifted byτ_(b) (340) in respect to revealing layer lenslet 301 b with center atposition 321. The base layer lenslet 302 c with center at position 325is shifted by τ_(c) (341) in respect to revealing layer lenslet 301 cwith center at position 322. When the base layer lenslet is in phase,illuminated region 365 a is observed by the observer as the color 355(green in the present example) sampled by the revealing layer lenslet301 a. When the base layer lenslets are out of phase, e.g. lenslet 302 bwith center 324 is at relative phase τ_(b)/T_(b) (340), a different partof the illuminated focal plane is sampled by the corresponding revealinglayer lenslet 301 b, here region 366 b of the focal plane, illuminatedby portion 356 of the illuminating light. As a further example, lenslet302 c is at phase τ_(c)/T_(b) and revealing layer lenslet 301 c samplesregion 367 c of the focal plane illuminated by portion 357 of theilluminating light. As illuminating light, light coming through a windowmay also be convenient, by having green portions from the trees and thegrass, gray and yellow portions from buildings and blue portions fromthe sky. Light reflected from a variable intensity and variable colorbackground such as a wall is also suitable. Light emitted by severalLEDs illuminating the lenslet setup from behind also provides excellentvisual effects. In addition, by using electronically driven multi-LEDdevices, i.e. several LEDS in a single package that emit differentcolors such as red, green and blue, one may create visually appealingcolors varying over time by pulse-width modulation of the individualcolored LEDs. With separate commands of the different packages of LEDs,one may obtain moiré level lines that in addition have smoothly evolvingcolors both spatially and over time.

Reinforcing the Contrast of the Base Layer by Diffusing Microlenses.

A light diffusing behavior in regions where the base layer lenslets areabsent (e.g. FIG. 4A, 402) can be achieved by placing small lenses 403randomly across the regions forming the background of the base layershape. These diffusing microlenses, line-oriented (1D) such as smallsegment cylindrical lenses or two-dimensional (2D) such as spherical oraspherical lenses should have a focal length significantly differentfrom the cylindrical lenslets 401 present in the base layer. By randomlypositioning and varying the size and therefore the focal length of thesediffusing microlenses, one creates a light diffusing effect thatconsiderably enhances the contrast between base layer shape foregroundregions and base layer shape background regions. As an example, FIG. 4Ashows the layout of the foreground regions 401 with the thin rectangularareas specifying the locations of the cylindrical lenses and the emptybackground regions 402. FIG. 4B shows the same views under a microscopewhere the cylindrical lenslets 404 form lenslet gratings that cover theforeground regions. Background regions are covered with the randomlypositioned diffusing microlenses 403. FIG. 4C shows an enlargement of apart of FIG. 4B.

Let us describe embodiments of the present invention for the 1D moirésynthesizing techniques, lenticular image synthesizing techniques, andfor the level line moiré synthesizing techniques.

Creating 1D Moiré with the Multi-Lenslet Imaging Setup

U.S. Pat. No. 7,710,551 (inventors Hersch and Chosson) discloses a “1Dmoiré image layout computation method” allowing the computation of thedirection and the speed in which 1D moiré image shapes move when therevealing layer samples, when tilting the setup, successive locations ofthe superposed base layer. Formula (1) to (5) describe according to U.S.Pat. No. 7,710,551 (inventors Hersch and Chosson) the mathematics usedfor computing the layouts of the base layer, given the layouts of therevealing layer and of the moiré layer.

Relying on the example of FIGS. 5A and 5B, let us first give therelationship between base band coordinates and moiré coordinates for arectilinear moiré, i.e. a moiré defined as a linear transformation ofthe replicated base bands. Base band 501 of base band period T_(b) withoblique base band letter shapes “VALIDE” is replicated by integermultiples of vector t=(t_(x), t_(y)) across the base layer to form thebase band grating. The corresponding moiré shapes 503 “VALIDE” areobtained by the revealing layer sampling lines 502 a, 502 b, 502 c, . .. having period T_(r) sampling the base bands successively at differentlocations. The vertical component t_(y) of base band replication vectort is equal to the base band period, i.e. t_(y)=T_(b). The moiré spacecoordinate (x,y) in function of the base space coordinates (x′,y′) is:

$\begin{matrix}{\begin{bmatrix}x \\y\end{bmatrix} = {\begin{bmatrix}1 & \frac{t_{x}}{T_{r} - T_{b}} \\0 & \frac{T_{r}}{T_{r} - T_{b}}\end{bmatrix}\begin{bmatrix}x^{\prime} \\y^{\prime}\end{bmatrix}}} & (1)\end{matrix}$

Equation (1), with matrix B=[1 t_(x)/T_(r)−T_(b); 0 T_(r)/T_(r)−T_(b)]expresses the linear relationship between base band space coordinates(x′,y′) and moiré space coordinates (x,y).

By inserting the components t_(x), t_(y) of base band replication vectort as (x′,y′) into Eq. (1), and equating t_(y)=T_(b), one obtains themoiré replication vector p=(p_(x), p_(y)). This calculation shows thatthe moiré replication vector p is the base band replication vector tmultiplied by T_(r)/(T_(r)−T_(b)).

The moiré height H is equal to the vertical component p_(y) of the moiréreplication vector p, i.e. H=p_(y). Therefore,

$\begin{matrix}{H = \frac{T_{r} \cdot T_{b}}{T_{r} - T_{b}}} & (2)\end{matrix}$

A designer can freely choose his moiré image height H and the directionof its movement α_(m) by defining replication vector p=(p_(x), p_(y)),with p_(y)=H and p_(x)=−H tan α_(m) and solve Eq. (1) for t using alsoEq. (2). This yields the base band replication vectort=p(T _(b) /H).  (3)

After selecting a suitable value for the revealing layer period T_(r),an imaging software module can then linearly transform a moiré imagedefined in the moiré coordinate space (x,y) into a base band defined inthe base layer coordinate space (x′,y′) by applying the inverse of Eq.(1), i.e.

$\begin{matrix}{\begin{bmatrix}x^{\prime} \\y^{\prime}\end{bmatrix} = {\begin{bmatrix}0 & \frac{T_{r} - T_{b}}{T_{r}} \\1 & {- \frac{t_{x}}{T_{r}}}\end{bmatrix}\begin{bmatrix}x \\y\end{bmatrix}}} & (4)\end{matrix}$

Let us now show how to generate a curvilinear moiré starting from arectilinear moiré. One may specify the layout of a desired curvilinear1D moiré image as well as the rectilinear or curvilinear layout of therevealing layer and the 1D moiré layout model is able to compute thelayout of the base layer.

The layout of the 1D moiré image in the transformed space is expressedby a geometric transformation M(x_(t),y_(t)) which maps the transformedmoiré space locations (x_(t),y_(t)) back to original moiré spacelocations (x,y). The layout of the revealing line grating in thetransformed space is expressed by a geometric transformationG(x_(t),y_(t)) which maps the transformed revealing layer spacelocations (x_(t),y_(t)) back into the original revealing layer spacelocations (x′,y′). The layout of the base band grating in thetransformed space is expressed by a geometric transformationH(x_(t),y_(t)) which maps the transformed base band grating locations(x_(t),y_(t)) back into the original base band grating locations(x′,y′). Transformation H(x_(t),y_(t)) is a function of thetransformations M(x_(t),y_(t)) and G(x_(t),y_(t)).

Let us define the transformations M, G, and H asM(x_(t),y_(t))=(m_(x)(x_(t),y_(t), m_(y)(x_(t),y_(t))),G(x_(t),y_(t))=(x, g_(y)(x_(t),y_(t)), andH(x_(t),y_(t))=(h_(x)(x_(t),y_(t), h_(y)(x_(t),y_(t)). According to thepublication by R. D. Hersch and S. Chosson, Band Moiré Images, Proc.SIGGRAPH 2004, ACM Trans. on Graphics, Vol. 23, No. 3, 239-248 (2004),the transformation of the moiré M(x_(t),y_(t)) is the following functionof the transformations of the base layer H(x_(t),y_(t)) and of therevealing layer G(x_(t),y_(t)):

$\begin{matrix}{{x = {{m_{x}\left( {x_{t},y_{t}} \right)} = {{h_{x}\left( {x_{t},y_{t}} \right)} + {\left( {{h_{y}\left( {x_{t},y_{t}} \right)} - {g_{y}\left( {x_{t},y_{t}} \right)}} \right) \cdot \frac{t_{x}}{T_{r} - t_{y}}}}}}{y = {{m_{y}\left( {x_{t},y_{t}} \right)} = {{{h_{y}\left( {x_{t},y_{t}} \right)} \cdot \frac{T_{r}}{T_{r} - t_{y}}} - {{g_{y}\left( {x_{t},y_{t}} \right)} \cdot \frac{t_{y}}{T_{r} - t_{y}}}}}}} & (5)\end{matrix}$where T_(r) is the period of the revealing line grating in the originalspace and where (t_(x), t_(y))=(t_(x), T_(b)) is the base bandreplication vector in the original space.

Then base layer transformation H(x_(t),y_(t)) is deduced from Eq. (5) asfollows when given the moiré layer transformation M(x_(t),y_(t)) and therevealing layer transformation G(x_(t),y_(t)) according to

$\begin{matrix}{{{h_{x}\left( {x_{t},y_{t}} \right)} = {{\left( {{g_{y}\left( {x_{t},y_{t}} \right)} - {m_{y}\left( {x_{t},y_{t}} \right)}} \right) \cdot \frac{t_{x}}{T_{r}}} + {m_{x}\left( {x_{t},y_{t}} \right)}}}{{h_{y}\left( {x_{t},y_{t}} \right)} = {{{g_{y}\left( {x_{t},y_{t}} \right)} \cdot \frac{T_{b}}{T_{r}}} + {{m_{y}\left( {x_{t},y_{t}} \right)} \cdot \frac{T_{r} - T_{b}}{T_{r}}}}}} & (6)\end{matrix}$Therefore, given the moiré layout and the revealing layer layout, oneobtains the backward transformation allowing to compute the base layerlayout. The moiré having the desired layout is then obtained by thesuperposition of the base and revealing layers.

Example A

Rectilinear 1D moiré image “EPFL” formed of revealing and base layerlenslets FIG. 6A shows schematically an example of a rectlinear moiréimage 603 formed of the superposition of base layer base bands 601 withforeground shapes 610 (black) and background shapes 611 (white) and arevealing layer 602 formed of an array of sampling lines 612. Therevealing layer period T_(r) is 400 μm, the base band period T_(b)=t_(y)is 364 μm and the moiré height is according to formula (2) H=4.044 mm.The transparent sampling lines 612 show the positions of the centerlineson which the cylindrical lenslets of the revealing layer are placed.FIG. 6B is an enlarged view of a part of FIG. 6A (compressed letter “F”)and shows the layout of the base layer rectangle array 615 for producingthe grating of cylindrical oblique lenslets forming the base layerforeground shapes 610. FIG. 6C shows a realization of the layout shownin FIG. 6B with a picture of the actual base layer lenslet grating 620laid out according to the layout 610 and along the obliquely laid outrectangle array 615. The base layer lenslet grating has a period ofT_(bl)=27 μm and a lenslet width W_(bl)=25 μm. The base layer backgoundshape 611 is populated with randomly placed 2D lenses 622 having a widthbetween 8 μm to 12 μm and focal lengths between 20 μm and 30 μm. Notethat the obliquely laid out base layer lenslet grating 620 createssharper moiré shapes.

FIG. 7A shows an alternative base layer layout for the small compressed“F” 610 of FIG. 6A. FIG. 7B shows a picture of a region 701 of thecorresponding implemented base layer lenslet grating seen under amicroscope. The foreground 610 is populated with a horizontally laid outlenslet grating, laid out according to the rectangle array 715 andrealized with the lenslet grating 720 (period T_(bl)=27 μm). Thebackground 711 is populated by a vertically laid out lenslet grating 721with a period of approximately one third the period of the horizontallylaid out foreground lenslet grating. Since the background lensletgrating has a focal length much smaller than the focal length of theforeground lenslet array, the light traversing it will be diffuse at thedepth of the focal plane of the revealing layer lenslet array. Incontrast, the light traversing the foreground lenslet grating will beconcentrated into the focal plane of the revealing layer lensletgrating. This statement is valid both for base layer backgound shapespopulated with small spherical lenslets or populated with smallcylindrical lenslets. The different light concentration behavior offoreground and background base layer lenslet gratings yields thecontrast that enables revealing the superposition images (1D moiré, 2Dmoiré, 2D random moiré, lenticular images).

FIG. 8 shows a photograph of a device formed of two lenslet layers, withthe “EPFL” moiré shape 802 formed of the superposition of thecylindrical base layer lenslets partly shown schematically in FIG. 7Aand of the cylindrical revealing layer lenslets 801. The revealing layerlenslet array has a lenslet period of 400 μm, with a lenslet width of385 μm and a gap of 15 μm between individual lenslets. FIG. 9 is aphotograph of the same device as in FIG. 8, but viewed from a slightlydifferent angle. The “EPFL” moiré shape 902 in FIG. 9 has movedvertically in respect to the “EPFL” moiré shape 802 shown in FIG. 8.

In order to prevent counterfeiters from creating duplicates of the baselayer lenslet foreground and background surfaces by surface duplicationmethods, and/or to protect the base layer lenslets against abrasure, onemay encapsulate it into another material having a lower index ofrefraction than the lenlets' material, e.g. an index of refraction of1.4 for a lenslet material of index of refraction of 1.5. Compared withlenslets in ambient air, the encapsulating material increases the focallength of the lenslets calculated according to formula (12), where theindex of refraction of the encapsulating material has to be inserted asn_(m).

Example B. Circular Band Moiré Image and Rectilinear Revealing Layer

In the present example, we choose a circularly laid out moiré image andalso freely choose the revealing layer layout. The desired referencecircular moiré image layout is given by the transformation mapping fromtransformed moiré space back into the original moiré space, i.e.

$\begin{matrix}{{{m_{x}\left( {x_{t},y_{t}} \right)} = {\frac{\pi - {{atan}\left( {{y_{t} - c_{y}},{x_{t} - c_{x}}} \right)}}{2\pi}w_{x}}}{{m_{y}\left( {x_{t},y_{t}} \right)} = {c_{m}\sqrt{\left( {x_{t} - c_{x}} \right)^{2} + \left( {y_{t} - c_{x}} \right)^{2}}}}} & (7)\end{matrix}$where constant c_(m) expresses a scaling factor, constants c_(x) andc_(y) give the center of the circular moiré image layout in thetransformed moiré space, w_(x) expresses the width of the originalrectilinear reference band moiré image and function a tan(y,x) returnsthe angle α of a radial line of slope y/x, with the returned angle α inthe range (−π<=α<=γ). The corresponding desired reference circular moiréimage is shown in FIG. 10A, 1003 and appears as the message “VALIDOFFICIAL DOCUMENT”. We take as revealing layer a rectilinear layoutidentical to the original rectilinear revealing layer, i.e.g_(y)(x_(t),y_(t))=y_(t) (1002). By inserting the curvilinear moiréimage layout equations (7) and the revealing layer layout equationg_(y)(x_(t),y_(t))=y_(t) into the band moiré layout model equations (6),one obtains the deduced curvilinear base layer layout equations

$\begin{matrix}{{{h_{x}\left( {x_{t},y_{t}} \right)} = {{\left( {y_{t} - {c_{m}\sqrt{\begin{matrix}{\left( {x_{t} - c_{x}} \right)^{2} +} \\\left( {y_{t} - c_{y}} \right)^{2}\end{matrix}}}} \right)\frac{t_{x}}{T_{r}}} + {\frac{\begin{matrix}{\pi - {atan}} \\\left( {{y_{t} - c_{y}},{x_{t} - c_{x}}} \right)\end{matrix}}{2\pi}w_{x}}}}{{h_{y}\left( {x_{t},y_{t}} \right)} = {{c_{m}\sqrt{\begin{matrix}{\left( {x_{t} - c_{x}} \right)^{2} +} \\\left( {y_{t} - c_{y}} \right)^{2}\end{matrix}}\frac{T_{r} - t_{y}}{T_{r}}} + {y_{t}\frac{t_{y}}{T_{r}}}}}} & (8)\end{matrix}$

These curvilinear base layer layout equations express the geometrictransformation from transformed base layer space to the original baselayer space. The corresponding curvilinear base layer in the transformedspace is shown in 1001. The resulting moiré image formed of thesuperposition of the base layer (1001) and of the revealing layer (1002)is shown in 1003. When the revealing layer 1002 is moved vertically overthe base layer 1001, the corresponding circular moiré image patternsmove radially and change their shape correspondingly. When movingtowards the exterior of the circular moiré, the letters become wider.

Let us consider example B embodied as a setup formed of two superposedlayers of lenslet gratings according to Section “Multi-lenslet imagingsetup with large revealing layer periods and small base layer periods”.FIG. 10B shows a photograph of a microscopic enlargement of part of thebase layer lenslets 1005 forming the base layer letters “ . . . UME . .. ” as embodiment of the lower central part of the base layer of FIG.10A (1001). When superposed with a grating of rectilinear revealinglayer sampling lines 1002, embodied by the cylindrical lenslet gratingappearing as slightly oblique lines in FIG. 10B (1006), the circularmoiré shape image appears. This moiré shape image is schematically shownin FIG. 10A (1003) and also shown as a photograph in FIG. 11 whenembodied as the setup formed of the superposed base and revealing layersof lenslet gratings.

Example C. Curvilinear Moiré Shape Image and Cosinusoidal RevealingLayer

Let us now take a curvilinear revealing layer and still generate thesame desired curvilinear moiré image as in the previous example B. Asexample, we take as curvilinear revealing layer a cosinusoidal layerwhose layout is obtained from the rectilinear revealing layer by acosinusoidal transformationg _(y)(x _(t) ,y _(t))=y _(t) +c ₁ cos(2 πx _(t) /c ₂)  (9)where constants c₁ and c₂ give respectively the amplitude and period ofthe cosinusoidal transformation. The corresponding cosinusoidalrevealing layer is shown in FIG. 12, 1202. By inserting the curvilinearmoiré image layout equations (7) and the curvilinear revealing layerlayout equation (9) into the band moiré layout model equations (6), oneobtains the deduced curvilinear base layer layout equations

$\begin{matrix}{{{h_{x}\left( {x_{t},x_{t}} \right)} = {{\left( {y_{t} + {c_{1}{\cos\left( \frac{2\pi\; x_{t}}{c_{2}} \right)}} - {c_{m}\sqrt{\left( {x_{t} - c_{x}} \right)^{2} + \left( {y_{t} - c_{y}} \right)^{2}}}} \right) \cdot \frac{t_{x}}{T_{r}}} + {\frac{\pi - {{atan}\left( {{y_{t} - c_{y}},{x_{t} - c_{x}}} \right)}}{2 \cdot \pi} \cdot w_{x}}}}{{h_{y}\left( {x_{t},x_{t}} \right)} = {{c_{m}{\sqrt{\left( {x_{t} - c_{x}} \right)^{2} + \left( {y_{t} - c_{y}} \right)^{2}} \cdot \frac{T_{r} - t_{y}}{T_{r}}}} + {\left( {y_{t} + {c_{1}{\cos\left( \frac{2\pi\; x_{t}}{c_{2}} \right)}}} \right) \cdot \frac{t_{y}}{T_{r}}}}}} & (10)\end{matrix}$

These curvilinear base layer layout equations express the geometrictransformation from the transformed base layer space to the originalbase layer space. The corresponding curvilinear base layer is show in1201. The superposition of the curvilinear base layer 1201 andcurvilinear revealing layer 1202 yields moiré image 1203. When therevealing layer 1202 is moved vertically over the base layer 1201, thecorresponding circular moiré image patterns move radially and changetheir shape correspondingly, as in the example shown in FIGS. 12 and 13.Let us consider example C embodied as a setup formed of two superposedlayers of lenslets according to Section “Multi-lenslet imaging setupwith large revealing layer periods and small base layer periods”. Thebase layer grating of lenses is placed at all foreground areas (“white”areas in FIGS. 10A, 12 and 13) of the base layer, in a similar manner asin FIG. 10B. These foreground areas act as a mask for the base layergratings of lenslets. The revealing layer grating of lenslets is formedof cylindrical lenslets following the cosinusoidal transparent (“white”in FIG. 12) lines 1202 of the revealing layer.

FIG. 13 shows that a slight displacement of the sampling locations ofthe revealing layer cylindrical lenslets obtained by a vertical shift ofthe revealing layer, or in case of a fixed setup obtained by tilting,yields a radial displacement of the revealed circular message. In FIG.13, the message “VALID OFFICIAL DOCUMENT” has moved radially in respectto its position in FIG. 12.

Dynamically Moving 1D and 2D Moiré Shapes Seen in Three DimensionsThanks to the Human Stereoscopic Vision

When creating a setup with vertically laid out revealing layer samplinglines, each eye of the observer sees at each position a slightlydifferent sampling of the base layer, i.e. each eye sees a slightlydifferent moiré image. Due to their slightly different sampling phase,these moiré images are slightly displaced one in respect to another andyield, thanks to human stereoscopic vision, an image having a certaindepth.

FIG. 14 shows schematically the geometry used for calculating the offsetbetween a moiré shape seen by the left eye (“L”) and a moiré shape seenby the right eye (“R”). This moiré offset, also called disparity,determines the apparent depth of the moiré shape. The space T_(E)between the left eye 145 and right eye 146 is generally around 63 mm. Weassume a viewing distance d (142, e.g. 360 mm) from the eyes to thesetup. The setup comprises a revealing layer 143 formed of a 1D gratingof lenticular lenslets of replication period T_(r) and of a base layer174 (see FIG. 17, showing the same view as FIG. 14, at a differentenlargement) formed of gratings of small cylindrical lenslets (FIG. 17,175). As shown in FIG. 14, the left eye (L) views the base layer acrossrevealing layer lenslet Λ₀ at position x_(0L)=x=0 and across lenslet Λ₁at position x_(1L). The right eye (R) views the base layer acrosslenslet Λ₀ at position x_(OR)=−γ_(b) and across lenslet Λ₁ at positionx_(1R).

The horizontal difference γ_(b) (difference along the x-axis), alsocalled base layer disparity, between the base layer spots x_(0L) andx_(OR) observed by the left and the right eyes respectively throughlenslet Λ₀ is according to the geometry of FIG. 14

$\begin{matrix}{\gamma_{b} = {\frac{T_{E}}{d + R_{r}}\left( {f_{r} - R_{r}} \right)}} & (11)\end{matrix}$where R_(r) is the lenslet curvature radius. For a circular lensletsection, curvature radius R_(r) is defined by the well-known relationgiving the focal length as a function of the curvature radius and theindex of refraction of the used material:

$\begin{matrix}{f_{r} = {R_{r}\frac{n_{lens}}{n_{lens} - n_{m}}}} & (12)\end{matrix}$where n_(lens) is the index of refraction of the lens material and n_(m)is the index of refraction of the surrounding medium, in the case ofair, n=1.

Through lens Λ₀ the right eye (R) views position

$\begin{matrix}{x_{0\; R} = {{- \gamma_{b}} = {{- \frac{T_{E}}{d + R_{r}}}\left( {f_{r} - R_{r}} \right)}}} & (13)\end{matrix}$

Through lenslet Λ₁ the right eye (R) views position x_(1R) which is

$\begin{matrix}{x_{1\; R} = {T_{r} - {\frac{T_{E} - T_{r}}{d + R_{r}}\left( {f_{r} - R_{r}} \right)}}} & (14)\end{matrix}$

Through lenslet Λ₂ the right eye (R) views position x_(2R) which is

$\begin{matrix}{x_{2\; R} = {{2T_{r}} - {\frac{T_{E} - {2T_{r}}}{d + R_{r}}\left( {f_{r} - R_{r}} \right)}}} & (15)\end{matrix}$

The positional x-increment from one revealing layer lenslet Λ_(i) to thenext Λ_(i+1) is therefore

$\begin{matrix}{{\Delta\; x_{r}} = {{T_{r} + {\frac{T_{r}}{d + R_{r}}\left( {f_{r} - R_{r}} \right)}} = {\frac{\begin{matrix}{{dT}_{r} + {T_{r}R_{r}} +} \\{{T_{r}f_{r}} - {T_{r}R_{r}}}\end{matrix}}{d + R_{r}} = {T_{r}\frac{d + f_{r}}{d + R_{r}}}}}} & (16)\end{matrix}$

This x-increment Δx_(r) is identical to the projected revealing layerperiod T_(r)′. According to FIG. 14, the projected revealing layerperiod T_(r)′, also called “apparent revealing layer period”, isobtained by projecting revealing layer period T_(r) from the plane 144connecting the central points c₀, c₁, c₂, . . . of the revealing layerlenses onto the focal plane 147 of the revealing layer lenses. Namely,

$\begin{matrix}{\frac{T_{r}^{\prime}}{T_{r}} = {\left. \frac{\left( {d + f_{r}} \right)}{d + R_{r}}\Rightarrow T_{r}^{\prime} \right. = {T_{r}\frac{\left( {d + f_{r}} \right)}{d + R_{r}}}}} & (17)\end{matrix}$

By comparing Eq. (16) and Eq. (17), one can easily verify that indeed,Δx_(r)=T_(r)′.

Let us now deduce the relative positions of the moiré shapes seen by theleft and right eyes. Thanks to stereoscopic fusion, the offset betweenthe moiré shapes seen by the left and right eyes yields the perceptionof depth.

FIG. 15 shows the setup from above (x-y plane), with separately the baselayer 151, the moiré shape seen by the left eye 152 and the moiré shapeseen by the right eye 153 for the case where the base layer band period(the period of repetition of the small “M” in FIG. 15) is smaller thanthe apparent revealing layer period, i.e. T_(r)′>T_(b). We assume thatthe left eye is in front of the setup, on the z-axis perpendicular tothe setup at position x=0. The vertical dashed lines x_(0L), x_(1L),x_(2L), x_(3L), x_(4L) . . . are the sampling lines of the revealinglayer lenslet grating when looked upon from the left eye. The verticaldashed lines x_(0R), x_(1R), x_(2R), x_(3R), x_(4R) . . . are thesampling lines of the revealing layer lenslets when looked upon from theright eye. By sampling the base layer shapes (here small letters “M”,each one inscribed within a base band), the sampling lines seen by theleft eye create the moiré shapes 152 that appear to the left eye and thesampling lines created by the right eye create the moiré shapes 153 thatappear to the right eye.

Let us now calculate the apparent height of the moiré seen by the leftor the right eye. The moiré shape is produced by the revealing layersampling lines projected onto the focal plane. The period of theseprojected revealing layer sampling lines is T_(r)′. According to Eq. (2)the apparent height H′ of the moiré is

$\begin{matrix}{H^{\prime} = \frac{T_{b} \cdot T_{r}^{\prime}}{T_{r}^{\;^{\prime}} - T_{b}}} & (18)\end{matrix}$

We can consider the moiré height H′ to be the moiré height apparent tothe eyes in the case of a revealing layer array of cylindrical lensletssampling a base layer formed of vertical base bands, where the base bandshapes are defined by small cylindrical lenslet gratings (e.g. 620 inFIG. 6C). Apparent moiré height H′ is calculated in the same manner asthe classical moiré height without gap between revealing layer and baselayer, but using as revealing layer period the revealing layer periodprojected onto the focal plane (Eq. (17)).

Let us now calculate the offset γ_(m) (also called disparity) betweenthe moiré shape seen by the left eye and the moiré shape seen by theright eye. Due to the position of the right eye (FIG. 14, 146), there isan offset between the base band locations x_(iL) sampled by therevealing layer lenslets when seen from the left eye and the base bandlocations x_(iR) sampled when seen from the right eye. At x_(0L)=0 forthe left eye, the corresponding offset x_(OR)=−γ_(b) for the right eyeis computed according to Eq. (11). It corresponds to a phase within therevealing layer period of ϕ_(r)=γ_(b)/T_(r)′. This means that byshifting the revealing layer by phase ϕ_(r), we bring the moiré seen bythe left eye and the moiré seen by the right eye onto the same position.Therefore, the offset γ_(m) between the moiré seen respectively by theleft and right eyes is the ratio ϕ_(r) of the apparent moiré height H′.We obtain

$\begin{matrix}{\gamma_{m} = {{\phi_{r}H^{\prime}} = {{\frac{\gamma_{b}}{T_{r}^{\prime}}\frac{T_{b} \cdot T_{r}^{\prime}}{\left( {T_{r}^{\prime} - T_{b}} \right)}} = \frac{\gamma_{b} \cdot T_{b}}{T_{r}^{\prime} - T_{b}}}}} & (19)\end{matrix}$

FIG. 16 shows the same elements as FIG. 15, with separately the baselayer 161, the moiré shape seen by the left eye 162 and the moiré shapeseen by the right eye 163 but for the case where the base layer bandperiod is larger than the revealing layer period, i.e. T_(b)>T_(r)′.Observe that in this case, the layout of the base layer shape (letter Mupright in positive x-direction) is inversed in respect to the layout ofthe moiré shape (letter upright in negative x-direction). This isexpressed in formula (18) by the negative apparent moiré height H′,resulting from the fact that T_(r)′−T_(b) is negative. Formula (19) alsoyields a negative offset γ_(m).

By inserting Eq. (11) into Eq. (19), we obtain the moiré offset γ_(m) asa function of the base layer period T_(b), projected revealing layerperiod T_(r)′, focal length f_(r) of the revealing layer lenslets andviewing distance d.

$\begin{matrix}{\gamma_{m} = {\frac{\frac{T_{E}}{d + R_{r}}{\left( {f_{r} - R_{r}} \right) \cdot T_{b}}}{T_{r}^{\prime} - T_{b}} = {\frac{T_{E} \cdot T_{b}}{\left( {T_{r}^{\prime} - T_{b}} \right)} \cdot \frac{\left( {f_{r} - R_{r}} \right)}{\left( {d + R_{r}} \right)}}}} & (20)\end{matrix}$

With the help of FIGS. 17 and 18, we can now calculate the apparentdepth of the moiré. The moiré shape offset γ_(m) 178 indicates theposition where the right eye, thanks to ray Q_(R), sees exactly the sameposition within the moiré shape as the left eye with ray Q_(L) atposition x=0. The intersection of rays Q_(R) and Q_(L) yields theapparent depth position D_(m) located at depth z=z_(m).

By considering the triangle formed of the left eye, right eye and depthposition D_(m), and the similar triangle formed by the origin of the x-zcoordinate plane (center of curvature C of the revealing layer lenslet),the intersection of ray Q_(R) with the x-axis and depth position D_(m),we obtain

$\begin{matrix}{\frac{z_{m}}{z_{m} + d + R_{r} + f_{r} - R_{r}} = {\frac{z_{m}}{z_{m} + d + f_{r}} = \frac{\gamma_{m}}{T_{E}}}} & (21)\end{matrix}$

Solving for the apparent moiré depth z_(m) yields

$\begin{matrix}{z_{m} = {\frac{\gamma_{m}\left( {d + f_{r}} \right)}{T_{E} - \gamma_{m}} = \frac{d + f_{r}}{\frac{T_{E}}{\gamma_{m}} - 1}}} & (22)\end{matrix}$

By inserting Eq. (20) into Eq. (22), we obtain for the apparent moirédepth

$\begin{matrix}\begin{matrix}{z_{m} = \frac{\gamma_{m}\left( {d + f_{r}} \right)}{T_{E} - \gamma_{m}}} \\{= \frac{{\frac{T_{E} \cdot T_{b}}{\left( {T_{r}^{\prime} - T_{b}} \right)} \cdot \frac{\left( {f_{r} - R_{r}} \right)}{\left( {d + R_{r}} \right)}}\left( {d + f_{r}} \right)}{T_{E} - {\frac{T_{E} \cdot T_{b}}{\left( {T_{r}^{\prime} - T_{b}} \right)} \cdot \frac{\left( {f_{r} - R_{r}} \right)}{\left( {d + R_{r}} \right)}}}} \\{= \frac{{T_{b}\left( {f_{r} - R_{r}} \right)}\left( {d + f_{r}} \right)}{{\left( {T_{r}^{\prime} - T_{b}} \right)\left( {d + R_{r}} \right)} - {T_{b}\left( {f_{r} - R_{r}} \right)}}}\end{matrix} & (23)\end{matrix}$

Since in the general case, the viewing distance d is large in respect tothe focal length, the simplified formula becomes

$\begin{matrix}\begin{matrix}{z_{m} = \frac{{T_{b}\left( {f_{r} - R_{r}} \right)}\left( {d + f_{r}} \right)}{{\left( {T_{r}^{\prime} - T_{b}} \right)\left( {d + R_{r}} \right)} - {T_{b}\left( {f_{r} - R_{r}} \right)}}} \\{\cong \frac{T_{b}\left( {f_{r} - R_{r}} \right)}{\left( {T_{r}^{\prime} - T_{b}} \right) - \frac{T_{b}\left( {f_{r} - R_{r}} \right)}{\left( {d + f_{r}} \right)}}} \\{\cong \frac{T_{b}\left( {f_{r} - R_{r}} \right)}{\left( {T_{r}^{\prime} - T_{b}} \right)}}\end{matrix} & (24)\end{matrix}$

When the apparent revealing layer period is smaller than the base layerperiod, i.e. T_(r)′<T_(b), according to Eq. (23) or Eq. (24), theapparent depth is negative and the resulting moiré shapes float in frontof the setup made of the two lenslet layers at a distance beingexpressed as a negative apparent depth value.

As an example, we consider a security design with two different moiré s,a first one with the “VALID” letters (FIG. 19A, 191 and FIG. 19B, 193)and a second one with the “OK” letters (FIG. 19A, 192 and FIG. 19B,194). The “OK” moiré moves in a direction opposite to the direction ofthe “VALID” moiré. The setup with the base and revealing layer lensletgratings is fixed on a glass plate. When tilting the setup horizontallyby a small angle, both the “OK” moiré shape and the “VALID” moiré shapemove in opposite directions, as shown by FIGS. 19A and 19B.

The moiré shapes shown in FIGS. 19A and 19B are also examples for stereomoiré vision. Let us first consider the first set of moiré shapes withthe “VALID” letters 191 or 193. Its parameters are the following: arevealing layer period T_(r)=0.4 mm, a base layer period T_(b)=0.353 mm,a focal length f_(r) of 1.2 mm, a viewing distance d=500 mm andaccording to Eq. (12) a radius of curvature R_(r)=0.4 mm. With Eq. (23)we obtain a calculated depth z_(m) of 6.01 mm, i.e. the moiré shapesviewed by superposed revealing and base lenslet layers have an apparentdepth of 6 mm. They seem to float behind the setup made of the twolenslet layers.

The second set of moiré shapes with the “OK” letters 192 or 194 has thesame parameters as the “VALID” moiré, but with a base layer periodT_(b)=0.446 mm, which is larger than the revealing layer periodT_(r)=0.4 mm. With Eq. (23) we obtain a calculated depth z_(m) of −7.79mm, i.e. the moiré shapes viewed by superposed revealing and baselenslet layers have an apparent depth of −7.8 mm. They seem to float infront of the setup made of the two lenslet layers.

Interestingly, in the general case, according to Eq. (23), when theviewing distance d is much larger than both the focal length f_(r) andthe base layer period T_(b), i.e., d>>f_(r) and d>>T_(b), the apparentdepth is largely independent of the viewing distance d. When the viewingdistance is changed for example in the range between 50 cm and 30 cm,the apparent depth remains substantially constant. In addition, when thebase layer period T_(b) comes closer to the apparent revealing layerperiod T_(r)′, the moiré sizes H and H′ increase and the apparent depthz_(m) also increases.

Creating 2D Moiré s with the Multi-Lenslet Imaging Setup

The theory regarding the analysis and synthesis of 2D moiré images isknown, see the following publications:

-   M. C. Hutley, R. Hunt, R. F. Stevens and P. Savander, “The moiré    magnifier”, Pure and applied Optics, Vol. 3, 133-142 (1994).-   H. Kamal, R. Völkel, J. Alda, Properties of the moiré magnifiers,    Optical Engineering, Vol. 37, No. 11, pp. 3007-3014 (1998).-   I. Amidror, The theory of the moiré phenomenon, Vol. 1, Section 4.4,    pp. 96-108 (2009)-   I. Amidror, R. D. Hersch, Fourier-based analysis and synthesis of    moiré s in the superposition of geometrically transformed periodic    structures, Journal of the Optical Society of America A, Vol. 15,    No. 5, May 1998, 1100-1113.

The sampling of a 2D array of micro-shapes (FIG. 20A, 2000) by an arrayof tiny holes 2001 or by a 2D array of microlenses yields 2D moiréshapes 2007 formed of enlarged and rotated instances of the micro-shape2003. We use here the formulation obtained by S. Chosson in his PhDthesis “Synthèse d'images moiré” (in English: Synthesis of moiréimages), EPFL Thesis 3434, 2006, pp. 111-112, referenced hereinafter as[Chosson 2006]. The denominations are similar as for the 1D moiré sdescribed in Section “Creating 1D moiré with the multi-lenslet imagingsetup”.

The layout of the 2D moiré image in the transformed space is expressedby a geometric transformation M(x_(t),y_(t)) which maps the transformedmoiré space locations (x_(t),y_(t)) back to original moiré spacelocations (x,y). The layout of the 2D revealing array in the transformedspace is expressed by a geometric transformation G(x_(t),y_(t)) whichmaps the transformed revealing array space locations (x_(t),y_(t)) backinto the original revealing layer array space locations (x′,y′). Thelayout of the 2D array of micro-shapes in the transformed space isexpressed by a geometric transformation H(x_(t),y_(t)) which maps thetransformed 2D micro-shape array locations (x_(t),y_(t)) back into theoriginal 2D micro-shape array locations (x′,y′).

A desired rectilinear or curvilinear 2D moiré image layout is specifiedby its moiré height H_(y) and width H_(x) in the original coordinatespace (x′,y′) and by its geometric transformation M(x_(t),y_(t)). Adesired revealing layer layout of the 2D sampling array is specified bythe period T_(rx) along the x-coordinate and T_(ry) along they-coordinate of its elements in the original space (x′,y′) and by itsgeometric transformation G(x_(t),y_(t)). The base layer layout of the 2Darray of micro-shapes is specified by the period T_(bx) along thex-coordinate and T_(by) along the y-coordinate of its elements in theoriginal space (x′,y′) and by its calculated geometric transformationH(x_(t),y_(t)). Having specified the desired 2D moiré image layout, thelayout of the 2D sampling revealing layer, and the size of themicro-shapes in the original space, then according to [Chosson 2006],the base layer geometric transformation H(x_(t),y_(t)) is obtained asfunction of the transformations M(x_(t),y_(t)) and G(x_(t),y_(t)).

Let us define the transformations M, G, and H asM(x_(t),y_(t))=(m_(x)(x_(t),y_(t), m_(y)(x_(t),y_(t))),G(x_(t),y_(t))=(g_(x)(x_(t),y_(t)), g_(y)(x_(t),y_(t)), andH(x_(t),y_(t))=(h_(x)(x_(t),y_(t), h_(y)(x_(t),y_(t))). Then, accordingto [Chosson 2006] transformation H(x_(t),y_(t)) is obtained by computing

$\begin{matrix}{{\frac{h_{x}\left( {x_{t},y_{t}} \right)}{T_{bx}} = {\frac{m_{x}\left( {x_{t},y_{t}} \right)}{H_{x}} + \frac{g_{x}\left( {x_{t},y_{t}} \right)}{T_{rx}}}}{\frac{h_{y}\left( {x_{t},y_{t}} \right)}{T_{by}} = {\frac{m_{y}\left( {x_{t},y_{t}} \right)}{H_{y}} + \frac{g_{y}\left( {x_{t},y_{t}} \right)}{T_{ry}}}}} & (25)\end{matrix}$

In the present invention, the revealing layer is embodied by a 2D arrayof lenslets, shown schematically by two lenslets in FIG. 21, 2105 andthe base layer by a 2D array of virtual micro-shapes shown schematicallyby two “$” signs 2103, created by having a 1D array of cylindricallenslets 2102 covering the foreground of each micro-shape. Note thateach microlens samples a different position within the virtualmicro-shapes of the base layer. For example, from a given observationposition, microlens 2116 samples position 2106 within the background ofthe micro-shape whereas microlens 2117 samples position 2107 within theforeground of the micro-shape. The background of the virtualmicro-shapes 2002 may be embodied by no lenses or by randomly locatedsmall microlenses 2022 diffusing the incoming light (see Section“Reinforcing the contrast of the base layer by diffusing microlenses”).

FIG. 20A shows the base layer 2000 and revealing layer 2001 auxiliarydigital images used to create the base layer embodied by arrays of smallsize 1D cylindrical lenslets (FIG. 20B, 2023) and the revealing layerembodied by 2D lenslets whose size is of the same order of size as thesize of the 2D base layer micro-shapes. FIG. 20B shows a photograph of amicroscopic view focussed on the base layer (“$” signs with foreground2023 and background 2022) with the revealing layer microlenses 2024appearing thanks to backlight illumination of the microscope. Theresulting 2D moiré shape 2007 represents the enlarged, rotated andsheared dollar sign 2003. The revealing layer lenslets of the 2D lensletgrating are centered at the holes 2006 of the revealing layer. In thebase layer, the gratings of small size 1D cylindrical lenslets 2023cover the foreground shapes 2003 of the virtual 2D array ofmicro-shapes. The background 2002 of the virtual micro-shapes is coveredby randomly placed microlenses 2022 having random sizes e.g. between 8μm to 12 μm, i.e. a diameter considerably smaller than the repetitionperiod of 27 μm of the lenslets forming the base layer lenslet gratings.

FIG. 22 shows a photograph of an embodiment of the 2D moirémulti-lenslet setup, consisting of a thin glass plate 2204. On the backside of this thin glass plate the base layer is pasted, which isembodied by 1D gratings of cylindrical lenslets yielding the virtualmicro-shapes. On the front side of the glass plate, the 2D revealinglayer lenslet grating 2205 is pasted. The resulting moiré foregroundshapes 2206 and moiré background shapes 2207 are clearly visible. Thedollar sign moves vertically when tilting the setup horizontally, i.e.rotating it slightly around a vertical axis and moves diagonally (at −45degrees) when tilting the setup vertically, i.e. rotating it slightlyaround a horizontal axis. The base layer 1D gratings of cylindricallenslets covering the foreground of the virtual microimages have alenslet repetition period of 16 μm. The revealing layer 2D lensletrepetition periods are 400 μm horizontally and vertically.

According to [Chosson 2006], for non-curvilinear moiré, i.e. forrectilinear moiré, the equation bringing moiré layer coordinates intobase layer coordinates by an affine transformation is the following:

$\begin{matrix}{\begin{bmatrix}x^{''} \\y^{''}\end{bmatrix} = {\frac{1}{{\left( {T_{rx} + v_{2x}} \right) \cdot \left( {T_{ry} + v_{1y}} \right)} - {v_{1x} \cdot v_{2\; y}}}{\quad{\left\lbrack \begin{matrix}{T_{rx} \cdot \left( {T_{ry} + v_{1y}} \right)} & {{- v_{1x}} \cdot T_{rx}} \\{{- v_{2y}} \cdot T_{ry}} & {T_{ry} \cdot \left( {T_{rx} + v_{2x}} \right)}\end{matrix} \right\rbrack\left\lbrack \begin{matrix}x \\y\end{matrix} \right\rbrack}}}} & (26)\end{matrix}$where {right arrow over (v)}₁=(v_(1z), v_(1y)) is defined as a firstmoiré displacement vector and {right arrow over (v)}₂=(v_(2x), v_(2y))is defined as a second displacement vector and where T_(rx) and T_(ry)are the revealing layer horizontal and vertical periods. As an example,FIG. 23 gives the coordinates of the desired moiré layout. The desiredmoiré displacement vectors are {right arrow over (v)}₁=(7500, −7500) and{right arrow over (v)}₂=(0, −10000). Inserting the coordinates of themoiré vertices A, B, C, D shown in FIG. 23 as (x,y) into Equation (26)yields the coordinates of the corresponding base layer vertices A″, B″,C″, D″ shown in FIG. 24. Therefore, for the two desired moirédisplacement vectors, and for given revealing layer periods, one maycalculate the base layer position x″, y″ corresponding to positions x, yin the moiré image. By inserting the moiré displacement vectors {rightarrow over (v)}₁ and {right arrow over (v)}₂ into Eq. (26), one obtainsthe corresponding base tile replication vectors {right arrow over (v)}₁″and {right arrow over (v)}₂″, see FIG. 24.

By inversion of formula (26), one obtains the affine transformationmapping base layer coordinates x″, y″ into moiré layer coordinates x, y:

$\begin{matrix}{\begin{bmatrix}x \\y\end{bmatrix} = {\begin{bmatrix}\frac{T_{rx} + v_{2x}}{T_{rx}} & \frac{v_{1x}}{T_{ry}} \\\frac{v_{2y}}{T_{rx}} & \frac{T_{ry} + v_{1\; y}}{T_{ry}}\end{bmatrix}\begin{bmatrix}x^{''} \\y^{''}\end{bmatrix}}} & (27)\end{matrix}$

By scanning the base layer (x″, y″) at successive x″ and y″ coordinates,scanline by scanline, the computer program finds according to Eq. (27)the corresponding locations x, y within the moiré image, reads at eachlocation the intensity or color and copies it back into the current baselayer location (x″, y″). This enables creating the corresponding baselayer 2D array of virtual micro-shapes. The foreground of these virtualmicro-shapes is then used as a mask for fabricating the 1D array ofcylindrical lenses.

Curvilinear moiré layouts described by a geometrical transformationM(x,y) may be produced by further applying the transformation H(x,y)described in Eq. (25) to the base layer array of virtual micro-shapes.

Level-Line Moiré s Embodied by the Multi-Lenslet Imaging Setup

U.S. Pat. No. 7,305,105 “Authentication of secure items by shape levellines” to Chosson and Hersch (also inventor in present invention), filedJun. 10, 2005, incorporated herein by reference, teaches how to create amoiré representing a freely chosen shape as successions of moiré levellines travelling from shape foreground and shape background skeletons tothe shape boundaries and vice-versa. The dynamically evolving levellines produced by a revealing layer grating sampling successivelocations of a base layer grating create the impression of a beatingshape.

Similar information as in U.S. Pat. No. 7,305,105 is presented in thepublication by S. Chosson and R. D. Hersch, Beating Shapes Relying onMoiré Level Lines, ACM Transactions on Graphics, Vol. 34, No. 1, Article9, 10 pages+two page Appendix, published in December 2014.

Level line moiré s rely on the principle stating that the level lines ofan elevation profile appear as moiré lines in the superposition of abase layer embodied by a line grating whose lines are shifted by anamount substantially proportional to the elevation and of a revealinglayer embodied by the unshifted line grating. We convert the bilevelshape that represents the outline of the desired moiré shape into anelevation profile. This elevation profile is conceived with the goal ofproducing strong intensity or color variations at the shape boundariesand of incorporating level lines that yield shapes similar to theoriginal bilevel shape.

The elevation profile level lines are revealed as moiré when superposingthe revealing line sampling grating on top of the synthesized base layerline grating incorporating the spatially dependent line shifts. Uponrelative displacement of the locations sampled by the revealing layer onthe base layer, the moving succession of moiré level lines creates theimpression of beating shapes.

As mentioned in Section “Multi-lenslet imaging setup with similarrevealing layer and base layer periods”, the revealing layer is embodiedby an array of cylindrical lenslets and the base layer is also embodiedby an array of cylindrical lenslets of a similar period, but shifted inrespect to the revealing layer lenslets according to the elevationprofile.

When an observer moves in respect to an illuminated multi-lensletimaging setup formed of the base and revealing layer lenslet gratings,level lines of the colors of the light sources move inwards and outwardsfrom the shape centers towards their boundaries and from the shapeboundaries towards the shape background centers.

A same geometric transformation applied to both the base and therevealing layers yields the same moiré shape that would be obtainedwithout geometric transformation. This enables creating cylindricallenslet arrays whose axes follow a spatial path given by a function,e.g. a cosinusoidal function defined by its period and amplitude.

By using a band-like dither array shifted according to the elevationprofile instead of simple shifted lines and by dithering an originalvariable intensity image, we create locally shifted base layer halftonelines of variable thickness embedding the elevation profile and at thesame time forming a halftoned instance of the original variableintensity image. To create halftone lines of variable thicknesses bycylindrical lenslets, one may cover the foreground area of the variablewidth halftone lines forming the base layer by oblique base layercylindrical lenslet gratings whose lenslets have a small repetitionperiod, in a similar manner as was carried out for 1D moiré shapes inSection “Creating 1D moiré with the multi-lenslet imaging setup”, seeExample A, FIGS. 6A, 6B, and 6C with base layer cylindrical lensletgrating 620.

With the known concept of indicial equations, we can deduce in a verysimple manner the curvilinear moiré fringes resulting from thesuperposition of a curvilinear base layer line grating e.g. embodied bya first base grating of cylindrical lenses and a possibly curvilinearrevealing layer line grating, e.g. embodied by a second revealinggrating of cylindrical lenses. The moiré fringes formed by thesuperposition of indexed line families form a new family of indexedlines whose equation is deduced from the equation of the base andrevealing layer line families, see the book by I. Amidror, The Theory ofthe Moiré Phenomenon, Vol. 1: Periodic Layers, 2^(nd) edition, section11.2, Springer, pp. 353-360 (2009). FIGS. 25A and 25B show the obliqueblack base layer lines with indices n=0, 1, 2, 3, . . . , thetransparent horizontal revealing layer lines with indices m=0, 1, 2, 3,4, . . . and the moiré fringe lines with indices k=−3, −2, −1, 0, 1.

The moiré fringe lines comprise dark moiré lines connecting theintersections of dark oblique and transparent horizontal revealing layerlines. As shown in FIGS. 25A and 25B, each dark moiré line ischaracterized by an index k which can be expressed by the subtraction ofthe revealing layer line index minus the base layer line indexk=m−n  (28)

The centerlines of the thick lines of the base layer form a line gratingparametrized by the integer values of the base layer line index n. Thisline grating is expressed byψ(x,y)=n T _(b)  (29)where ψ(x,y)=0 expresses the implicit equation of either a straight orof a curvilinear line and where T_(b) defines the line period. Forexample, in the case of a straight line grating of orientation θ as inFIG. 25A we havey cos θ−x sin θ=n·T _(b)  (30)where T_(b) is the perpendicular distance between successive lines.

In the general case, the revealing line grating is expressed byΨ(x,y)=m T _(r)  (31)where Ψ(x,y) expresses the implicit equation of the revealing layerlines in the target space and where T_(r) is the period of thecorresponding rectilinear horizontal revealing line grating in theoriginal space. For example, a horizontal revealing line grating isexpressed byy=m T _(r)  (32)

Thanks to equation (28), and by expressing indices n and m according toEqs. (29) and (31) as functions of x and y, the implicit equation of themoiré fringe lines becomes

$\begin{matrix}{{\frac{\Phi\left( {x,y} \right)}{T_{r}} - \frac{\Psi\left( {x,y} \right)}{T_{b}}} = k} & (33)\end{matrix}$

For example, in the case of the superposition of the oblique rectilinearbase layer grating having angle θ and of a horizontal revealing linegrating as shown in FIG. 25A, the moiré fringe line Eq. (33) becomes

$\begin{matrix}{{\frac{y}{T_{r}} - \frac{{y\;\cos\;\theta} - {x\;\sin\;\theta}}{T_{b}}} = k} & (34)\end{matrix}$and, by rearrangingy·(T _(h) −T _(r) cos θ)+x·T _(r) sin θ=k·T _(r) ·T _(b)  (35)

Equation (35) fully describes the family of moiré fringe lines (FIG.25A, 2501). Integer values of k correspond to the centerlines of the“thick lines” forming the moiré fringe lines and real values of kcorrespond to lines located within bands whose boundaries are formed ofthe moiré center lines.

Let us describe in more details the level line moiré. Level line moiré senable visualizing the level lines of an elevation function G(x,y) bysuperposing a base layer grating whose horizontal lines are verticallyshifted according to the elevation function G(x,y) and a horizontalrevealing layer grating having the same line period as the base layergrating. We consider the case where both the revealing layer grating andthe base layer grating have the same period, i.e. T=T_(r)=T_(b).

The base layer grating is described by the line familyy−G(x,y)=n·T  (36)

With a horizontal revealing line grating y=m T of the same period T asthe base layer grating, we obtain according to Eq. (33) the equation ofthe moiré fringe lines

$\begin{matrix}{{\frac{y}{T} - \frac{y - {G\left( {x,y} \right)}}{T}} = {\left. k\Rightarrow{{G\left( {x,y} \right)}/T} \right. = k}} & (37)\end{matrix}$

Therefore, the revealed moiré fringe lines form the level lines ofelevation function G(x,y).

Let us consider non-linear geometrical transformations applied to boththe base and revealing layer line gratings. For example, FIG. 25B showsthe result of applying different non-linear geometrical transformationsto the gratings of FIG. 25A. The moiré lines can still be indexed byk=m−n and Eq. (33) describing the resulting moiré layout remains valid.Here also we consider the case where both the revealing layer gratingand the base layer grating have the same period, i.e. T=T_(r)=T_(b).

We consider a geometric transformation y′=Q(x,y) mapping the targetspace (x,y) containing the curvilinear base and revealing line gratingsback into the original space (x′,y′) containing the rectilinearhorizontal base and revealing line gratings. Since the originaluntransformed rectilinear base and revealing line gratings arehorizontal, the geometric transformation is completely defined byy′=Q(x,y).

We obtain the revealing layer's curvilinear line grating in the targetspace by traversing all discrete pixel locations (x,y) of the targetspace, finding their corresponding locations (x′=x, y′=Q(x,y)) in theoriginal space, obtaining their intensities, respectively colors andaccordingly, setting the intensities, respectively colors, of thecorresponding target space pixels. We obtain the base layer'scurvilinear line grating in the target space in a similar manner byapplying the geometric transformation to obtain original space locations(x′=x, y′=Q(x,y)), then locating the shifted positions y′−G(x,y),obtaining their intensities, respectively colors and setting accordinglythe intensities, respectively colors, of the corresponding target spacepixels.

By applying the geometric transformations to the revealing and baselayers, we obtain their respective layouts Ψ(x,y)=Q(x,y) andw(x,y)=Q(x,y)−G(x,y). Inserting these layouts into Eq. (32), yields themoiré line family

$\begin{matrix}{{\frac{Q\left( {x,y} \right)}{T} - \frac{{Q\left( {x,y} \right)} - {G\left( {x,y} \right)}}{T}} = {\left. k\Rightarrow\frac{G\left( {x,y} \right)}{T} \right. = k}} & (38)\end{matrix}$

Eq. (38) shows that when applying a same geometric transformation to thebase and the revealing layers, one obtains as moiré fringes the levellines of elevation function G(x,y). Geometric transformations compriseseveral freely choosable parameters, which can be used as keys toconstruct many different pairs of matching base and revealing layergratings. This is important for document security applications.

Construction of Level Line Moiré s

In order to produce a level line moiré, we start with a bilevel shape asshown in FIG. 26A. By computing the skeletons of the foreground andbackground shapes (see G. G. Sanniti Di Baja, E. Thiel, Skeletonizationalgorithm running on path based distance maps, Image and VisionComputing Vol. 14, 47-57, 1996), we obtain an intermediate skeletonrepresentation, FIG. 26B of the bilevel image. Then, by a distancetransform, we establish the relative distance d_(krel) of each pixel(x,y) in the interval between the shape boundary 2610 and its respectiveskeleton (foreground skeleton 2611 or background skeleton 2612), i.e.,d_(krel) expresses the relative distance of pixel (x,y) to itsrespective skeleton on a scale between 0 and 1. We then map the relativedistances D_(krel) onto elevations. Clearly visible moiré shapes with ahigh gradient or a discontinuity at their shape boundary are obtained byassigning to foreground shapes the elevations between 0.5 and 1 and tobackground shapes the elevations between 0 and 0.5. In more generalterms, foreground and background elevation values areh _(ƒ)(x,y)=h _(fs)−ƒ(d _(krel)(x,y))(h _(fs) −h _(fc)), andh _(b)(x,y)=h _(bs)−ƒ(d _(krel)(x,y))(h _(bs) −h _(bc))  (39)respectively, where h_(fs) and h_(bs) are the elevation values of theforeground and background skeletons respectively, and where h_(fc), andh_(bc) are the elevation values at the foreground and background shapeboundaries, respectively. Function ƒ(d_(krel)(x,y))=d_(krel)(x,y)^(γ)provides either directly the relative distance (γ=1) or a power functionof the relative distance between a point and its skeleton. By applying asubsequent optional low-pass filtering step, the elevationdiscontinuities at the shape boundaries can be smoothed out. This helpsin making the local line grating shifts induced by the elevation profileless visible.

In order to illustrate the synthesis of level line moiré, we use a 1Dgrating of bands as base layer. Each band is formed of an intensitygradient (FIG. 27, 2701) perpendicular to the band orientation. At eachlocation, this grating of bands is vertically shifted in proportion tothe elevation at the corresponding location. The resulting shifted baselayer is shown in (FIG. 27A). A one period maximal shift corresponds tothe maximum of the elevation profile. In order to obtain a fasterdisplacement of the moiré, one may choose to have a maximal shiftcorresponding to one and a half, two or more periods of the base layerline grating. By superposing the sampling revealing layer grating on topof the shifted base layer line grating, one obtains the moiré shapesformed by the level lines of the elevation profile (FIG. 27B).

In one embodiment, the base layer is formed of a 1D grating ofcylindrical lenslets centered on the shifted bands having substantiallythe same period as the unshifted 1D grating of cylindrical lensletsforming the revealing layer. FIG. 28A shows a photograph of anembodiment of the level line moiré multi-lenslet setup. This setupconsists of a thin glass plate 2804 on whose back side the base layerembodied by the 1D grating of partially shifted cylindrical lenslets ispasted and on whose front side the non shifted 1D grating of cylindricallenslets 2805 is pasted. The resulting moiré level lines 2806 areclearly visible as constant intensity lines located between theforeground or background shape skeletons and the moiré shape boundaries.

FIG. 28B shows an enlargement of a portion 2808 of the moiré shape imageof FIG. 28A, which shows that a high intensity 2810 appears at positionswhere the base layer grating of cylindrical lenses concentrates theincoming light at the positions sampled by the revealing layer gratingof cylindrical lenses, i.e. along the viewing direction.

FIGS. 29A, 29B, 29C and 29D show each the front lenslet 2920,respectively 2950 representative for the revealing layer grating ofcylindrical lenslets in front of the eye 2900 and the back lenslet 2921,respectively 2951 representative for the base layer gratings ofcylindrical lenslets behind the revealing layer grating.

For a point light source 2905 located perpendicularly behind themulti-lenslet setup 2910 formed of base and revealing layers of lensletgratings and an observer 2900 viewing the multi-lenslet along its normal(FIG. 29A), the intensity is the highest (FIG. 28B, 2810) when the twogratings 2920 and 2921 are in phase 2901. A lower intensity (FIG. 28B2811) appears at positions where the base 2950 and revealing 2951 layergratings cylindrical lenses are shifted one in respect to another. Atpositions of the setup 2911 where the two gratings are out of phase,i.e. one grating is shifted 2902 by approximately half a period (φ) inrespect to the other grating, the resulting intensity (FIG. 28B 2812) isthe lowest.

But, as shown in FIG. 29C, when the setup 2910 formed of the superposedgratings of cylindrical lenses is tilted in respect to the viewer and tothe light source, when the two gratings are in phase 2903, the baselayer lenslet concentrates the incoming light 2905 at a position 2930different from the sampling positions 2931 of the revealing layerlenslet and the corresponding locations are dark. As shown in FIG. 29D,at positions where the two gratings are shifted 2904 by a certainfraction ϕ of the grating period, the light concentrated by the baselayer lenslet 2940 is sampled by the revealing layer lenslet 2941 andprovides a high intensity to the viewer 2900. These relative shiftsbetween the two gratings of cylindrical lenslets are responsible for thedifferent intensities of the different level lines in the level linemoiré shapes shown in FIG. 28A.

Level Line Moiré with a Geometric Transformation of the Gratings

One may also apply a geometric transformation to both the base andrevealing layer gratings, before shifting the base layer gratingaccording to the elevation profile. As an example, consider thetransformation y′=Q(x,y) mapping the geometrically transformed targetplane locations (x,y) back into the non-transformed plane (x′,y′)y′=Q(x,y)=y+c ₁ cos(2π(x+c ₃)/c ₂)  (40)where c₁, c₂, and c₃ are parameters of the cosinusoidal transformation.By inserting the cosinusoidal transformation expressed by Equation (40)into the moiré fringe layout equation (10), we obtain the equation ofthe moiré line family

$\begin{matrix}{{\frac{y + {c_{1}{\cos\left( {2{{\pi\left( {x + c_{3}} \right)}/c_{2}}} \right)}}}{T} - \frac{y + {c_{1}{\cos\left( {2{{\pi\left( {x + c_{3}} \right)}/c_{2}}} \right)}} - {G\left( {x,y} \right)}}{T}} = {\left. k\Rightarrow{{G\left( {x,y} \right)}/T} \right. = k}} & (41)\end{matrix}$i.e., the moiré is formed by the level lines G(x,y)/T=k. This means thatwe obtain the same level line moiré as the one we would obtain withoutgeometric transformation.

By being able to freely choose the transformation parameters c₁, c₂, andc₃, we can create a variety of different transformations. Only arevealing layer grating matching the set of parameters of the base layergrating will be able to correctly reveal the hidden level line moiré. Asan example, FIG. 30A shows a geometrically transformed base layergrating formed of an array of base layer bands each one containing anintensity gradient having a triangular shape. The geometrictransformation is a cosinusoidal transformation as defined by Eq. (40).The intensity gradient perpendicular to the band orientation (seeenlargement 3001) is representative for the light concentrated by thebase layer lenslet gratings on the focal plane of the revealing layergrating. This gradient intensity band is vertically shifted by an amountsubstantially proportional to the elevation profile of the motives to berevealed as moiré level line shapes, here the letter “A” and the “heart”motives, see the upper half of FIG. 26C. When superposed with a samplinggrating of revealing layer cylindrical lenslets laid out along thecenter of the transparent lines (white lines in FIG. 30B), thecorresponding level line moiré shape appears (FIG. 30C). By slightlydisplacing the sampling positions of the revealing layer lenslet gratingon its focal plane, i.e. in respect to the base layer lightconcentrating lenslet gratings, a different instance of the same levelline moiré shape appears (FIG. 30D). These sampling locations aredisplaced when viewing from a different angle the setup formed of thebase and revealing layer lenslet gratings, see Section “Visible effectobtained by the level line moiré”.

One may also apply a geometric transformation to both the base andrevealing layer gratings, after having shifted the base layer gratingaccording to the elevation profile. In this case, the level line moiréis also geometrically transformed and may become curvilinear. As anexample, see U.S. Pat. No. 7,305,105 to Chosson and Hersch, column 14,lines 25 to 65. FIGS. 19 and 20 in U.S. Pat. No. 7,305,105 show theresulting geometrically transformed level line moiré. By replacing thecurvilinear revealing layer lines of FIG. 18 in U.S. Pat. No. 7,305,105by a curvilinear grating of cylindrical lenslets following the whitelines and by filling the white shape areas of FIG. 19 in U.S. Pat. No.7,305,105 with small obliquely oriented lenslet gratings as in FIG. 34Aof the present application, one achieves a similar level line moiré asthe one shown in FIGS. 19 and 20 of U.S. Pat. No. 7,305,105, butrealized with a setup of base layer and revealing layer gratings made oflenslets.

Level Line Moiré Representing Grayscale Images

By using as elevation profile a grayscale image such as a human face,one may then reveal as level line moiré the level lines of the face.With a multi-lenslet setup made of a base layer grating of cylindricallenslets which are shifted according to the face intensities and arevealing layer grating of unshifted cylindrical lenslets, one may viewat a certain orientation of the setup the human face where the cheeksare bright (e.g. FIG. 31A, 3101) and the hair is dark 3102 or byslightly tilting the setup, the cheeks become dark (FIG. 31B, 3111) andthe hair is bright 3112.

The setup shown in the examples of FIGS. 31A and 31B is a concreteembodiment made with gratings of cylindrical lenslets having anindividual lenslet period of 50 μm. Depending on the application,cylindrical lenslet periods of a few microns to centimeters arepossible.

Level Line Moiré Produced with a Revealing Layer Lenslet Grating ofLarge Repetition Period and Base Layer Lenslet Gratings of SmallRepetition Period Forming a Halftone Image

A further variant of creating level line moiré s by base layer lensletsforming a halftone image consists in creating from an original image(e.g. FIG. 32A) a halftone image (e.g. FIG. 32B) with substantiallyparallelogram shaped black halftone screen elements (enlargement ofrectangular area 3201 in FIG. 33F) embedding the elevation profile(s)(e.g. FIGS. 33A and 33B) of the message to be revealed as level linemoiré shape. This is carried out by creating the base layer halftonelenslet array as follows.

(A) Create a first array of dither bands oriented at a first angle θ₁(e.g. 60°) having a gray intensity gradient with values between 0 and 1,with the dither bands shifted according to a previously prepared firstelevation profile (FIG. 33C).

(B) Create a second array of dither bands oriented at a second angle θ₂(e.g. −60° having a gray intensity gradient with values between 0 and 1,with the dither bands shifted according to a previously prepared secondelevation profile (FIG. 33D).

(C) Combine the values of the first and the second dither arrays term byterm by taking the minimum value, and apply a histogram equalizationprocedure. The resulting dual band dither array shows two intersectinggratings of bands (FIG. 33E).

(D) Halftone an input grayscale image (e.g. FIG. 32A) by dithering withthe dual band dither array resulting from step (C). The resultinghalftone elements will consist of black quadrilaterals and white areas(FIG. 32B with enlargement in FIG. 33F). The black quadrilateralhalftone areas have a straight or slightly deformed parallelogram shape,well suited for the placement of the small period lenslet arrays.(E) Place the lenticular base layer lenslet gratings represented byarrays of longitudinal rectangles (FIG. 34B, 3402) within eachquadrilateral black halftone element area (FIG. 33F), preferablyaccording to the orientation of one of the dither bands (e.g. the ditherband orientation of FIG. 33C, also present in FIGS. 33F and 34A). Thiscan be done by intersecting a large array of rectangles representing thelayout of the lenslets at the selected orientation with the blackhalftone areas ((black areas in FIG. 33F). Lenslet rectangle partsoutside the black halftone area are eliminated.(F) Use the longitudinal rectangles laid out in step E (FIG. 34A) toexpose the resist used for fabricating the base layer comprising thelenslet gratings.

In the present multi-lenslet grating embodiment of the level line moiré,the dither band repetition period is substantially larger than therepetition period of the lenslets forming the base layer lensletgratings. The revealing layer lenslet grating period is the same as thedither band repetition period. The present embodiment enables creating,with base layer lenslet gratings, a halftone image such as the one shownin FIG. 32B. The quasi-parallelogram shape of the black halftoneelements facilitates the placement of the base layer lenslets andenables, when superposing a revealing layer lenslet grating at a firstorientation (3502), to reveal a first hidden message 3505 in oneorientation (see FIG. 35A corresponding to the rectangular surface 3202in FIG. 32B) and when superposing a second revealing layer 3503 at asecond orientation, to reveal a similar or a different hidden message inthe second orientation (FIG. 35B, 3506). The halftone layer 3501 shownin FIGS. 35A and 35B corresponds to rectangular area 3202 in FIG. 32B.

The solution shown above relies on small period lenslet gratings for theblack screen element parts and no lenslets for the black screen elementparts. The light through the lenslets gets diffused whereas lightthrough the areas without lenslets has a strong intensity. Othervariations are possible, for example by replacing the blackparallelogram parts containing the cylindrical lenslet arrays withrandomly placed light diffusing microlenses, as described in Section“Reinforcing the contrast of the base layer by diffusing microlenses”.

The present embodiment has the advantage of showing a halftone image,e.g. the face of a known person or the face of the document holder andat the same time being able to reveal, when superposed with therevealing layer lenslet grating, additional information, such as thename and birth date of the person whose face appears as halftone image.This solution is specially promising when using digital systems that areable to print personalized lenslet grating layouts. In addition, it ispossible to replace the revealing layer sampling lenslet grating by asmartphone, tablet or laptop computer programmed to acquire the halftoneimage formed by the base layer lenslet gratings, to perform by softwarean “AND” operation between the acquired halftone base layer image and acorresponding revealing layer array of transparent lines located inmemory, to show the resulting additional information on the displayand/or to recognize the information content by optical recognitionsoftware and to send the resulting information content to a server forvalidation (similar to FIG. 46 and Section “Verification of theauthenticity of a security feature relying on layers with superposedgratings of lenslets”, third embodiment).

Visible Effect Obtained by the Level Line Moiré

When moving the revealing layer lenslet grating in respect to the baselayer lenslet grating(s) or in the case of a fixed setup of base andrevealing layer lenslet gratings, when tilting the setup in respect tothe viewer, the constant intensity or color lines representing the levellines of the elevation profile incorporated into the base layer byshifts of its lenslet gratings appear to move between the elevationshape foreground skeleton and the shape boundaries and between theelevation shape boundaries and the shape background skeleton. In FIGS.30C and 30D, the level lines are represented by constant gray values.The different sampling positions of the revealing layer in respect tothe base layer shown in FIGS. 30C and 30D create a different mappingbetween the level lines of the elevation profile and theirrepresentation as constant intensity or constant color lines. Theconstant intensities or colors seen as level lines depend on thebackside illumination that should vary according to the incident angleon the base layer lenslets (see incident light in FIG. 3, at angularorientations 354, 355, 356, 357).

Lenticular Image Synthesizing Techniques

The lenticular image synthesizing techniques rely on base layer slicing,sampling, compression and re-assembling and on revealing layer sampling.A number of input images, e.g. the 4 images shown in FIG. 36, boxes A,B, C and D are sliced and compressed by a factor of 4, then superposed,yielding the base layer assembly shown in box E. Individual images 3601,3602, 3603, 3604 become image parts 3605, 3606, 3607, 3608,respectively. Slices 3611, 3612, 3613 and 3614 become after compression(scaling down in one direction) slices 3615, 3616, 3617 and 3618,respectively. In order to realize a setup made of base layer andrevealing layer gratings of lenslets, the black foreground slices shownin box E are embodied by gratings of oblique cylindrical lenslets. Thebackground (white) is left empty or filled with small microlensesdiffusing the light. The revealing layer grating of lenslets haslenslets with the same period as the distance 3620 between successiveslices of a same image part.

As an example, the FIGS. 37A, 37B, 37C, and 37D show photographs of asetup made of base and revealing grating lenslets. This setup shows assuperposition shapes four successive views representing a running rabbitthat appear when tilting the setup vertically. The revealing layercylindrical lenslets are horizontally oriented and have a period of 400μm, with each cylindrical lens having a width of 385 μm followed by agap of 15 μm. The base layer gratings covering the foreground of thebase layer images are formed of cylindrical lenslets having a period of16 μm, with each cylindrical lens element having a base width of 14 μmand a gap of 2 μm. The base layer gratings are rotated by 15 degrees inrespect to the orientation of the revealing layer grating.

Further Superposition Images Created by Superposed Gratings of Lenslets

The lenslet grating synthesizing techniques described in detail abovethat create upon observation 1D moiré shapes, 2D moiré shapes, levelline moiré shapes or lenticular image shapes are only a subset of thevariety of superposition shapes that can be achieved with superposedlayers of lenslet gratings. These lenslet grating synthesizingtechniques can also be used to create in an analogous manner a pair ofbase and revealing layer random 2D lenslet gratings that show bysuperposition a single instance of a 2D moiré view, according to U.S.Pat. No. 7,058,202 to Amidror. They can also be used to create a pair ofbase and revealing layer random 1D lenslet gratings that show a singleinstance of a 1D moiré shape, according to U.S. Pat. No. 8,351,087 toAmidror and Hersch.

Fabrication of Gratings of Cylindrical Lenslets

The technologies developed to produce arrays of spherical microlensesare also applicable to produce gratings of cylindrical lenslets. Thefollowing list of publications gives an overview about possibletechnologies for fabricating microlenses.

-   a) Z. D. Popovic, R. A. Sprague, and G. A. N. Connell, “Technique    for monolithic fabrication of microlens arrays,” Applied Optics,    vol. 27, no. 7, p. 1281, April 1988.-   b) D. Daly, R. F. Stevens, M. C. Hutley, and N. Davies, “The    manufacture of microlenses by melting photoresist,” Measurement    Science and Technology, Vol. 1, 759-766 (1990)-   c) C. Y. Chang, S. Y. Yang, M.-H. Chu, Rapid fabrication of    ultraviolet-cured polymer microlens arrays by soft roller stamping    process, Microelectronic Engineering, Vol. 84, 355-361 (2007).-   d) C. N. Hu, H. T. Hsieh, G. D. J. Su, Fabrication of microlens    arrays by a rolling process with soft polydimethylsiloxane molds, J.    of Micromechanics and Microengineering, Vol. 21, paper 065013, 7 pp.    (2011).-   e) S. J. Liu, C. C. Huang, C. T. Liao, Continuous Infrared-Assisted    Double-Sided Roll-toRoll Embossing of Flexible Polymer Substrates,    Polymer Engineering and Science, Vo. 52, Issue 7, 1395-1401, (2012).

Let us first present one embodiment based on the reflow of positiveresist. The gratings of cylindrical lenslets are fabricated by directlaser writing and reflow of positive resists spin coated on silicon. Thearrays are replicated in PDMS and finally used for UV imprint withphotocurable polymer. At the end of the process, a sixth step can beexecuted for the encapsulation of the device. The different steps areshown as cross-sections in FIG. 38.

-   A. Positive resist 3801 is spin coated on a silicon wafer (3800).-   B. The resist is exposed and developed (3810) so as to yield    longitudinal stripes 3811 following the centerlines of the    cylindrical lenslets.-   C. The structured resist is heated and becomes liquid. This reflow    process tends to create, due to surface tension, cylindrical    lenslets whose cross sections are circular segments (3812).-   D. In order to create the molds, Polydimethylsiloxane (PDMS, e.g.    Sylgard 184) 3813 mixed with a curing agent is placed on top of the    reflowed resist structures. It is heated to around 80° C. during at    least one hour so as to fully polymerize. The PDMS is removed from    the silicon substrate and now holds the negative shape of the    cylindrical lenslets.-   E. The PDMS is then placed on top of a suitable transparent    substrate (glass or plastic sheet 3824) on which UV curable material    3814 such as Ormocomp was deposited. The UV curable material spreads    out under the PDMS. It is then exposed from below or from the top by    a UV light source 3815. The PDMS mold is released and the grating of    cylindrical lenslets is available, bound to its transparent    substrate.-   F. Option: In order to provide a higher resistance and a longer    lifetime, the grating of cylindrical lenslets can be encapsulated by    a transparent material such as a polymer 3816 having a lower index    of refraction than the index of refraction of the lenslets. For    example, one may achieve this with a lenslet material of index    n=1.52 and an encapsulating polymer of index n=1.4. The radius of    the cylindrical lenslets is computed according to formula (12) to    yield the desired focal length.

Let us now describe embodiments enabling the mass production of gratingsof cylindrical lenslets. A first embodiment relies on a roll-to-rollsetup with a single PDMS carrying wheel.

Points A to D are the same as above, but carried out on a lengthysurface comprising several parts (areas) each one having its gratings ofcylindrical lenslets. In addition, according to FIGS. 38 and 39, thefollowing steps are carried out.

-   E′. The PDMS molds (FIG. 39, 3903) are attached to a belt 3902    encircling the rolling wheel. The substrate 3910 formed e.g. of    plastic is pressed against the PDMS encircling the rolling wheel    3901 by a secondary pressing wheel 3905. UV curable material such as    Ormocomp is continuously deposited 3904 onto the substrate before    passing through the two wheels.-   F. Then, both the PDMS mold and the pressed curable material travel    with the rotation 3907 of the main wheel 3901. During this travel,    the curable material is pressed into the PDMS mold and at the same    time it is cured by UV illumination 3908.-   G. When the molded cured material exits 3906 the rotation of the    main wheel, the PDMS follows the surface of the main wheel and the    molded cured material separates from it. The molded cured material    is available as gratings of lenslets 3909 on the substrate 3910.

Either steps A to E or alternately, steps A to G can be performed bothfor the base layer gratings of cylindrical lenslets and for therevealing layer gratings of cylindrical lenslets. In order to enhancethe contrast in the base layer (see Section “Reinforcing the contrast ofthe base layer by diffusing microlenses”), microlenses can be formed ina similar manner as cylindrical lenslets, by also exposing in step Bcircular disks, as known in the art.

The final multi-lenslet setup containing the base and revealing layergratings of lenslets can be obtained by pasting the base layer gratingand revealing layer grating together. Registration marks formed of across made of 2 cylindrical lenslets located between the separate partsmay be helpful for bringing the two layers into registration beforepasting them together.

A second roll-to-roll embodiment consists in imprinting both layers ofthe substrate on the same installation, as shown in FIG. 40. This systemrelies on one wheel 4001 carrying a first PDMS mold 4011 for imprintingthe base layer lenslets and on a second wheel 4002 carrying a secondPDMS mold 4013 for imprinting the revealing layer lenslets. On each sideof the substrate the system works in the same manner as described forthe single PDMS carrying wheel embodiment described previously. Notethat optionally, after being imprinted by passing along the first wheel,the material can be encapsulated by a transparent layer of a lower indexof refraction, yielding a flat surface. In step G, after the moldedcured material 4012 separates from the PDMS, curable material is poored4060 on the other side of the substrate and is pressed against thesecond PDMS mold 4013 while rotating along the second wheel 4002. Thematerial is cured by UV light 4021, hardened and finally detached fromthe PDMS 4014. As a result of this process, the base layer grating oflenslets 4051 on one side of the substrate is available simultaneouslywith the revealing layer grating of lenslets on the other side of thesubstrate 4014. In order to ensure a good spatial registration of thetwo layers of lenslets, registration marks formed of a cross made of 2small cylindrical lenslets are placed at regular intervals between theseparate areas of the gratings. When setting up this process, the secondwheel 4002 can be slightly moved in respect to the first wheel 4001 soas to ensure that the hardened “cross” registration marks made oflenslets are brought into registration at the output of the secondwheel.

Other embodiments are possible, for example, an embodiment relying ondouble side roll-to-roll embossing of flexible polymer substrates byrollers bearing the negative of the desired lenslet gratings. One rollerbears the negative of the base layer grating of lenslets and the othermetallic roller bears the negative of the revealing layer grating oflenslets. The two metallic rollers are pressed together and the curablematerial is poored on both sides of a glass or plastic substrate. Bypressure between the two wheels, the curable material takes the desiredshape and by UV illumination the material is cured. In case oftransparent rollers, the UV light sources may illuminate the curablematerial from inside the wheels.

Fabrication of Personalized Gratings of Lenslets

High anti-counterfeiting security is provided by individualized gratingsof lenslets that comprise an information related to the specificdocument or valuable article that is to be protected. For example, therevealing and base layer gratings of lenslets may be affixed on a boxcontaining valuable drugs, where the superposition image shows aslevel-line moiré the number characterizing the production series of thatdrug or its specific identification number. As a further example, an IDcard may show as base layer lenslet gratings the halftone image of thecard holder and as superposition moiré shape, either as a 1D movingmoiré shape or as a level line moiré shape, representing the birth dateof the card holder. The production of such personalized gratings oflenslets can be obtained by printing with a polymer jetting printer(FIG. 41, 4100) working in a similar manner as an inkjet printer. Thethin polymer stripes, rectangles 4130 or circular disks are printed andthen, due to surface tension, possibly upon heating 4106 and/or UVcuring 4107, form the cylindrical or circular lenslet gratings,respectively. For more details about possible polymer jetting processes,see the following articles, herein incorporated by reference:

-   Fakhfouri et al., “Inkjet printing of SU-8 polymer based mems, a    case study for microlenses”, Proc. IEEE 21st International    Conference on Micro Electro Mechanical Systems, 2008;-   Vilmi et al., “Inkjet printed micro lens array on patterned    substrate”, SPIE Vol. 8613, article no. 861317, 2013;-   Chen et al., Fabrication of inkjet-printed SU-8 photoresist    microlenses using hydrophilic confinement, J. Micromech. Microeng.    Vol. 23, article 065008, 8 pages (2013).

The recently developed 3D printers may also print such lenses withpolymer material that at a certain temperature form, due to surfacetension, the desired cylindrical or circular lenslet gratings. Thepolymerization can then proceed by irradiating the resulting lensletgratings with UV light and/or by heating them.

Large size lenslet gratings, e.g. lenslet gratings with a repetitionperiod larger than 1 mm, may be manufactured by classical 3D printingwith a transparent plastic material. The 3D shape of the one sided or incase of a fixed setup of the two-sided lenslet gratings is described bya surface model such as the STL file format for stereolithography. TheSTL description is then entered into the software converting the surfacemodel to printer commands specifying the x-y horizontal displacementsand the z vertical displacements of the print head. The resultingprinted 3D plastic element forms either the base layer lenslet gratingsor the revealing layer lenslet grating on top of a flat layer ofplastic. In case that both the base and revealing layer gratings areprinted at the same time on the two side of a flat layer of plastic, theresulting printed 3D volume is the fixed multi-lenslet setup directlyusable to view the resulting superposition shape image. Such a fixedsetup of large size, from a few centimeters to several meters, can beused for advertisement and decoration, in exhibitions, for thedecoration of walls or in amusement parks.

A digital fabrication line producing personalized security devices suchas identity cards (“ID cards”) comprises a computer (FIG. 41, 4110)running a computer program that automatically produces the layout of the1D arrays of rectangular areas and 2D arrays of circular areas on whichthe corresponding cylindrical and spherical lenslet gratings are formed.

In the case of a level line moiré showing as base layer the halftoneimage e.g. of the document holder and as level line moiré his name andbirth date, the computer program may carry out the following steps:

-   A) Read the record from disk 4112 or from a network server 4111    containing the variable intensity (or grayscale) face image and the    name and birth date of the document holder;-   B) Create the elevation profile(s) (e.g. FIGS. 33A and 33B) from the    bitmap image of the birth date;-   C) Create the dither array (e.g. FIG. 33E) with the band arrays    (FIGS. 33C and 33D) incorporating by appropriate shifts the    elevation profile(s) created in point B above;-   D) Halftone the variable intensity face image (e.g. 32A) by    dithering with the dither array created in point C above, thereby    yielding a halftone image with black polygonal halftone element    surfaces (e.g. FIGS. 32B and 33F);-   E) Fill the quadrilateral black halftone surfaces with arrays of    small rectangles (FIG. 34A and enlargement in FIG. 34B) defining the    layout of the cylindrical lenslet gratings;-   F) With the array of small base layer rectangles, form the commands    (4104 print head commands and table displacement commands 4115 and    4116) for the polymer jetting printer that prints the polymer    material 4103 on the selected substrate 4102 (e.g. plastic). These    commands may directly be commands to print successive droplets along    these rectangles or the raster-scan printing of polymer droplets    according the rasterized rectangle array file. The produced    substrate with the polymer droplets is then heated 4106 and/or UV    cured 4107.

If the revealing layer lenslet grating is not individually personalized,it may be fabricated as described in Section “Production of gratings ofcylindrical lenslets”. If it is personalized, for example by having forpersonalized security items cosinusoidal revealing layer layouts ofdifferent amplitudes, periods and orientations, it may be produced in amanner analogous to the procedure described above for creatingpersonalized base layer lenslet gratings.

The fabrication of large sized fixed setups of lenslet gratings wouldcomprise the step of conceiving the fixed setup of base and revealinglayer lenslet gratings (FIG. 42, 4201), of deriving 4202 from theparameters and layout of the lenslet gratings a 3D surface description(e.g. STL), of translating the 3D surface description into 3D printinghead displacement commands 4203 and of printing 4204 the fixed setup in3D according to the computed head displacement commands.

Conceiving a Security Feature

The method for conceiving a security feature (FIG. 43) that is to beincorporated into a document or an article to be secured comprises thefollowing steps.

-   -   a) Select a category of superposition effects 4301, such as        lenticular image effect, 1D moiré s, 2D moiré s, random moiré,        level line moiré s, shift effects or lenticular imaging effects.    -   b) Conceive the message 4302 that should appear as superposition        shape image.    -   c) Run the layer synthesizing software 4303 operable for        creating the selected category of superposition effects and give        as input the message, as well as appropriate parameters (see the        relevant sections describing the different superposition        effects).    -   d) Obtain as output (i) the base layer grating layout in the        form of a bitmap 4304 showing the portions (foreground) that are        to be filled with the base layer gratings of lenslets and (ii)        the revealing layer layout in the form of a bitmap with        transparent lines or tiny holes specifying the layout of the        revealing layer grating of lenslets.    -   e) Verify that the superposition of the base and revealing        layers yields the desired superposition shape image        incorporating the initially conceived message.    -   f) Apply a further processing step in order to fill the        foreground portions of the base layer with surfaces representing        the layout of the base layer gratings of lenslets 4305. Choose        an appropriate orientation and period of the base layer lenslet        gratings.    -   g) Depending on the input files required by the equipment        operable for exposing the resist in step A of section        “Production of gratings of cylindrical lenslets”, an additional        processing step may be required to produce the files defining        the path of an exposing laser. For example, the stripes that        form the surfaces of the resist which are later treated by the        reflow process are converted into sequences of small CIF        (Caltech Intermediate Form) rectangles. The exposing laser then        exposes the successive rectangles. The output of the present        step is the CIF file acting as input to the laser exposure        software.    -   h) Finally, fabricate the lenslet gratings 4306.        Placement of the Recto-Verso Gratings of Lenslets onto Valuable        Documents and Products

The multi-lenslet setup incorporating on its recto the revealing layergrating of lenslets and on its verso the base layer gratings of lensletsprovides the clearest superposition shape image when viewed intransmission mode, when light arrives from the back of the setup, e.g.light from a window, light from an artificial light source, light froman array of LEDs or light from a wall. The best effects are achievedwhen the incoming light varies in intensity according to its incidentangle. The setup with the recto-verso gratings of lenslets can beincorporated into any document window, e.g. the window reserved fortransmissive effects on opaque paper banknotes, the non-opacifiedportion of a polymer banknote or a transparent polymer area within aplastic card. The recto-verso gratings of lenslets can also be placed ona transparent portion of the polymeric data page of a passport.Recto-verso gratings of lenslets can also be easily incorporated intotransparent or semi-transparent areas of plastic indentity cards (IDcards) by placing them on both sides of the card. They can beencapsulated by a transparent material such as a polymer with a lowerindex of refraction than the index of refraction of the lenslets. Theresulting ID card may for example show the face of the ID card holder.By tilting the ID card, the face parts change smoothly their intensitiesfrom highlight to dark and vice versa, as shown in FIGS. 31A and 31B.

According to Section “Level line moiré produced with a revealing layerlenslet grating of large repetition period and base layer lensletgratings of a small repetition period forming a halftone image”, it isalso possible to print the base layer lenslet gratings forming thehalftone image on one side of a semi-transparent substrate and therevealing layer lenslet grating on the other side of thesemi-transparent substrate. Then, when viewed in reflective mode on adark background (FIG. 44A, 4401), the halftone image 4402 is visible,e.g. an image resembling the photograph of the document holder, and whenviewed in transmissive mode 4403, illuminated from behind 4405, thetransmissive superposition image formed of the superposition of the baseand revealing layer gratings becomes visible. This transmissivesuperposition image contains a message such as the name, birth date andID number of the document holder 4404.

The setup with the recto-verso gratings of lenslets can also be appliedon any package reserving a transparent window for this authenticationfeature. For example, a package containing drugs may incorporate a smalltransparent window located in its pivoting lid. This transparent windowmay incorporate on one side the revealing layer grating of lenslets andon the other side the light concentrating gratings of lenslets formingthe base layer. When opening the box, the lid shows as moirésuperposition image the dynamically moving “ORIGINAL DRUG” message.

Packages that include a transparent part or a transparent window arevery often used for selling a large variety of products, including, forexample, CDs, DVDs, etc., where the transparent part of the packageenables customers to see the product inside the package. The transparentparts of such packages may also be used advantageously forauthentication and anti-counterfeiting of the products, by using a partof the transparent window for the placement of the recto-verso gratingsof lenslets. The setup made of the recto-verso gratings of lenslets mayalso be printed on separate security labels or stickers that are affixedor otherwise attached to the product itself or to the package.

Verification of the Authenticity of a Security Feature Relying on Layerswith Superposed Gratings of Lenslets

In one embodiment of the present invention, the shape image resultingfrom the superposition of the base and revealing layer gratings oflenslets can be visualized by simply looking at the setup incorporatingthe layers of lenslets. This superposition shape image may represent agraphic motif, a symbol or a piece of text that is known to characterizethe item that is to be authenticated. By modifying the relative samplingposition of the revealing layer grating of lenslets in respect to thebase layer grating of lenslets concentrating the incoming light, thesuperposition shape image becomes animated. The relative samplingposition of the revealing layer grating can be modified e.g. by arelative translation or a relative rotation of the layers or by tiltinghorizontally, vertically, or diagonally a fixed setup (e.g. FIG. 45A andFIG. 45B, 4503) formed of the two layers. In case of a 1D or 2D moiré,the superposition shape image moves (e.g. from 4505 to 4506) across thesetup (see also evolution of the moiré shape image from FIG. 12 to FIG.13). In case of a level line moiré, the superposition shape image showsbeatings from the foreground and background centers to the borders ofthat shape and vice-versa (see FIGS. 30C and 30D for two differentsampling positions; the constant gray levels follow the shape levellines). In case of a lenticular image, several shape instances give theimpression of a dynamically moving shape (see FIGS. 37A, 37B, 37C and37D).

In a second embodiment, several superposition image messages may beincorporated into the same setup of base and revealing gratings oflenslets. For example, FIGS. 19A and 19B show the 1D moiré messages“VALID” and “OK” that appear at different depth levels and move inopposite directions. A person can immediately verify with the naked eyethe authenticity of the security item by tilting the fixed setup andverifying that the two messages move in opposite directions and that the“VALID” message appears more distant than the “OK” message.

In a third embodiment, the shape image (FIG. 46, 4610) resulting fromthe superposition of the base and revealing layer gratings of lensletscan be photographed by a smartphone 4604. The software of the smartphonecan analyze that shape image, e.g. in case of a text message comprisingcharacters and/or numbers, it can recognize the message 4605 by opticalcharacter recognition software and possibly interact 4621 with a Webserver 4620 in order to verify whether this identifying text message isvalid. If the identifying text message is valid, a message 4606 appearson the smartphone telling the observer that the security itemincorporating the superposed base and revealing grating of lenslets isauthentic.

Anti-Counterfeiting Features

Without appropriate sophisticated equipment capable of performing thelithography (or laser exposure) and the reflow operations, it is notpossible to replicate the base and revealing layer gratings of lenslets.Even if such an equipment is available to the potential counterfeiter,attempts to falsify a secure item produced in accordance with thepresent invention by taking microscope images of the grating of lensletswill slightly change the size of the corresponding lenslets. The moiréshapes that are obtained with the 1D moiré, 2D moiré and the level linemoiré s are very sensitive to the ratio between revealing layer and baselayer lenslet periods. Small changes of these ratios may create verylarge distortions of the resulting superposition shape images. Inaddition, the cylindrical revealing layer grating of lenslets may have acurved layout such as a cosinusoidal layout. Without knowing theparameters of the corresponding geometric transformation, such curvedrevealing layer gratings would be very difficult to counterfeit.Finally, either one or both the base layer gratings of lenslets and therevealing layer grating of lenslets may be encapsulated by a transparentmaterial such as a polymer having a lower index of refraction than thelenslets. Such an encapsulation makes it very difficult for acounterfeiter to recover by imaging means the orientation, size andlayout of the lenslet gratings.

Decorative Aspects

In addition to security, the presented setups of revealing and baselayer lenslet gratings have a high esthetical value and may also beattractive in luxury products such as watches, smartphones, perfumes,expensive drinks, in clothes such as dress, skirt, blouse, jacket,shawls and pants as well as in bikes and cars (see also U.S. Pat. No.7,295,717, incorporated by reference, where one of the inventors is thesame as in the present invention). In addition, due to their unexpectedappearance and the dynamicity of the resulting superposition shapeimage, these setups may also be created at a large scale for exhibitionsor for amusement parks. They also may find applications for thedecoration of buildings. At these large scales, base and revealing layergratings of lenslets may be created by filling plastic cylinders orspherical elements with a liquid such as water to obtain cylindricallenslets or spherical lenslets.

The invention claimed is:
 1. A method for creating a superposablerevealing layer lenslet grating and at least one base layer lensletgrating yielding, when superposed, a superposition shape image thatshows a recognizable message, comprising the steps of: (i) selecting therecognizable message that is to appear as superposition shape image;(ii) selecting a layer layout synthesizing technique yielding saidsuperposition shape image; (iii) defining layer layout parametersaccording to the selected layer layout synthesizing technique; (iv)generating from the shape image of the selected recognizable messageboth the revealing and base layers, where the revealing layer comprisesspatial information specifying the layout of the revealing layer lensletgrating and where the base layer comprises spatial informationspecifying the layout of the base layer lenslet gratings; (v)fabricating the revealing layer lenslet grating laid out according tothe revealing layer spatial information and the base layer lensletgratings laid out according to the base layer spatial information byapplying techniques selected from the set of lithographic techniques,laser writing techniques, etching techniques, reflow techniques andembossing techniques; where said revealing layer lenslet grating andbase layer grating comprise arrays of cylindrical lenses; where saidrecognizable message is selected from the set of text, numbers,graphical symbols, picture with recognizable elements, face image,typographical characters, numerals, logos, and spatial codes; where thesuperposition shape image results from the sampling action of therevealing layer lenslet grating on the plane on which the base layerlenslet grating concentrates the incoming light; and where upon tiltingof the superposed revealing and base layer gratings the superpositionshape image evolves dynamically.
 2. The method of claim 1, where thesuperposed base and revealing layer lenslet gratings form a fixedmulti-lenslet setup, where said layer layout synthesizing technique is alevel-line moiré synthesizing technique, where the setup is illuminatedfrom behind and where the dynamic evolution is characterized by constantintensity or color lines travelling across successive level lines of ashape elevation profile, said level lines being located between thesuperposition shape boundaries and the superposition shape foregroundand background centers.
 3. The method of claim 1, where the step offabricating the revealing layer lenslet grating and the base layerlenslet gratings comprises also fabricating at least one additionallayer encapsulating one of the lenslet gratings, where saidencapsulating layer has an index of refraction lower than the index ofrefraction of the lenslet gratings, and where said encapsulating layerhas a flat interface with the air hiding the layout of the encapsulatedgrating, thereby preventing its replication for counterfeiting purposes.4. The method of claim 2, where a specific geometrical transformationfrom transformed space to original space is applied to the revealinglayer and where, (a) for a level line moiré having the same appearanceas the level line moiré created with a rectilinear revealing layer, thebase layer is generated according to the same specific geometrictransformation as the revealing layer and the shape elevation profile isincorporated into the base layer by vertical shifts in the originalspace, said vertical shifts being a function of the profile elevation;(b) for a curvilinear level line moiré being geometrically transformedaccording to the same geometric transformation as the revealing layer,the elevation profile is incorporated in the base layer by verticalshifts in the transformed space, said vertical shifts being a functionof the profile elevation.
 5. The method of claim 1, where the spatialinformation specifying the layout of the revealing layer lenslet gratingis an array of revealing layer transparent lines, where the spatialinformation specifying the layout of the base layer lenslet gratings isselected from the set of arrays of base layer transparent lines, arraysof rectangles and arrays of disks, where in case of base layertransparent lines, the fabricated base layer lenslet grating hassubstantially the same period as the fabricated revealing layer lensletgrating and where in case of arrays of rectangles and arrays of disks,the fabricated lenslet gratings have a substantially smaller periodcompared with the period of the revealing layer lenslet grating.
 6. Themethod of claim 5, where the base layer has a foreground filled withsaid arrays of base layer rectangles and a background filled withrandomly positioned non-overlapping disks of sizes that are randomlyselected within a given size interval and are substantially smaller thanthe period of the revealing layer grating and where during thefabrication step, cylindrical lenslet gratings are created at locationsof said array of base layer rectangles, and microlenses are created atthe positions of the randomly positioned non-overlapping disks.
 7. Themethod of claim 1, where the superposed base and revealing layer lensletgratings form a fixed setup, where the revealing layer lenslet gratinghas a substantially vertical orientation, thereby providing to the eyesof an observer different views of the base layer lenslet gratings, saiddifferent views creating a parallax effect allowing to perceive thesuperposition shape image as an image composed of shapes havingdifferent apparent depths.
 8. The method of claim 7, whose superpositionshape image is composed of a first message and of a second message,where when tilting the setup, the first message moves at a givenapparent depth level and the second message moves in inverse directionat a different apparent depth level.
 9. The method of claim 2, where thesetup is illuminated from behind by an illumination providing spatiallyvarying colors, said illumination creating level lines that have colorssimilar to the colors present in the illumination.
 10. The method ofclaim 9, where the illumination is formed of several light emittingdiode packages placed at different positions behind the setup, and wherethe level lines have similar colors as the light from the light emittingdiode packages.
 11. The method of claim 10, where the colors emitted bythe light emitting diode packages evolve over time, and where thereforethe level lines have colors that also evolve over time.
 12. The methodof claim 9, where, when the setup is tilted with respect to an observer,the color level lines evolve both spatially and over time.